Tilburg centre for Creative Computing
P.O. Box 90153
Tilburg University
5000 LE Tilburg, The Netherlands
http://www.uvt.nl/ticc
Email: ticc@uvt.nl
Copyright © S.M. Dodunekov, A. Bojilov and A.J. van Zanten 2011.
February 14, 2011
TiCC TR 2011-001
Generalized Residue Codes
and their Idempotent Generators
Bulgarian Academy of Sciences, Bulgaria and
TiCC, Tilburg University
S.M. Dodunekov, A. Bojilov and A.J. van Zanten
!
Generalized Residue Codes
and their
Idempotent Generators
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#$%&'()'*!+,'-./0!12!3,).*,.45!#$%&'()'!'*-!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!6)775!6)%8$(&!9*):.(4);0!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!3<=<!>1-$*.?1:5!+<!#1@)%1:!'*-!+<A<!:'*!B'*;.*!
!
"
Abstract
D.*.('%)E.-!F.4)-$.!G!"#H,1-.4! ( '$ 5& 5% !'(.!,0,%),!,1-.4!12!%.*&;I! ' !1:.(!;I.!2).%-! !) G&H 5!
J);I! G'5 &H ! " 5!JI)%.! % !I'4!;1!8.!'!-):)41(!12! " G'H 5!JI.(.! " !)4!;I.!K$%.(!2$*,;)1*<!6I.!
L'('/.;.(! $ !,'*!I':.! % !-)22.(.*;!:'%$.4!21(!2)M.-!:'%$.4!12! ' 5! & !'*-! % 5!'*-!;I.! % !
,1((.4L1*-)*&!!"#,1-.4!'(.!'%%!.N$):'%.*;<!O*!;I)4!F.L1(;!J.!4;$-0!;I.!)-./L1;.*;!
&.*.(';1(4!12!4$,I!,1-.4<!O*!L'(;),$%'(5!J.!4I'%%!-1!;I)4!21(! ' $ P *5 C *5 * # 5 C * # Q 5! # % " 5!
JI.*!;I.!,1-.4! ('$ 5 & 5% !8.%1*&!;1!;I.!4$8,%'44!12!%#(.4)-$.!,1-.4<!KML%),);!.ML(.44)1*4!'(.!
-.():.-!JI),I!!.ML(.44!;I.!)-./L1;.*;!&.*.(';1(4!)*!;.(/4!12!;I.!4)E.4!12!,0,%1;1/),!
,14.;4!/1-! ' !J);I!(.4L.,;!;1!&5!'*-!12!;I.!,1.22),).*;4!12!;I.!1*.!8$;!I)&I.4;!L1J.(!12! + !
)*!;I.!)((.-$,)8%.!L1%0*1/)'%4!1:.(! !) G& H JI),I!-):)-.! + ' & " <!
!
C
Contents
"<!O*;(1-$,;)1*!'*-!-.2)*);)1*4!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!L<!S!
!
C<!O-./L1;.*;!&.*.(';1(4!12!,0,%),!,1-.4!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!L<!T!
!
R<!+!/';()M!-.;.(/)*)*&!'%%!L()/);):.!)-./L1;.*;!&.*.(';1(4!!!!!!!!!!!!!!!L<!"U!
!
S<!O-./L1;.*;!&.*.(';1(4!12!!"#,1-.4!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!L<!CV!
!
V<!6I.!,'4.! ' ! * ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,L<!CT!
,
W<!6I.!,'4.! ' ! C * !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!L<!R"!!!
,
U<!6I.!,'4.! ' ! * # !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!L<!RW!
,
T<!6I.!,'4.! ' ! C * # !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!L<!SS!
!
X<!+!2.J!(./'(?4!1*!;I.!$*(.4;(),;.-!,'4.4! ' ! * # !'*-! ' ! C * # !!!!!!!!!!!!!!L<!V"!
!
"Y<!6I.!!"#,1-.4!12!%.*&;I!C"!1:.(! !) GSH !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!L<!VU!
!
F.2.(.*,.4!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!L<!WX!
!
R
1. Introduction and definitions
O*!Z"[!&.*.('%)E.-!(.4)-$.!,1-.4!'(.!-.2)*.-!'4!21%%1J4<!D):.*!41/.!L14);):.!)*;.&.(!'!
'*-!41/.!L()/.!L1J.(!&!J);I! G'5 &H ! " <!\.;! - ]! !1(- ' G& H '*-!%.;!.!8.!;I.!/$%;)L%),';):.!
&(1$L!&.*.(';.-!80!&/,)<.<! . ]! P"5 &5<<<<<<<5 & - &"Q <!^.!'%41!)*;(1-$,.!;I.!&(1$L!!]_ 0 ' 5!
8.)*&!;I.!&(1$L!12!L14);):.!)*;.&.(4!/1-!'/,JI),I!'(.!L()/.!;1!'1!6I.*!! 0 ' !!_!! " G'H !
'*-!.!)4!'!4$8&(1$L!12! 0 ' <!!`$(;I.(/1(.5!);!)4!J.%%!?*1J*!G,2<!'%41!Z"5G"<CH[H!;I';!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! + ' & " ! G + & "H 2G +H' ' G +H 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!G"H!
!
JI.(.! ' ' G +H !)4!;I.! '%3 !,0,%1;1/),!L1%0*1/)'%!)*! !) G&HZ +[ !12!-.&(..! " G'H 5!'*-!JI.(.!
2G +H !)4!;I.!L(1-$,;!12!'%%!,0,%1;1/),!L1%0*1/)'%4! ' 4 G + H !J);I!4,!'5!! " ( 4 ( ' <!^.!'%41!
?*1J!;I';! ' ' G +H !,'*!8.!2',;1()E.-!)*;1! ) ]! " G'H a - !)((.-$,)8%.!L1%0*1/)'%4! 2$ G + H 5!
" * $ * ) 5!1:.(! !) G&H !
!
)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ' ' G +H ! + 2$ G +H <!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GCH!
$ !"
b.M;5!J.!,I114.!'!&(1$L!5!5!4$,I!;I';!!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! . , 5 , ! <!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GRH!
!
6I.!)*-.M!12!5!J);I!(.4L.,;!;1!!!)4!,'%%.-!%5!JI)%.!;I.!)*-.M!12!.!J);I!(.4L.,;!;1!5!)4!
,'%%.-!61!6I.!,14.;4!12!5,J);I!(.4L.,;!;1!!,'(.!-.*1;.-!80! 5" ]! 5 5 5 C 5<<<<<<<<5 5 % 5!'*-!;I.!
,14.;4!12!.,J);I!(.4L.,;!;1!5,80! P." ]! . 5 . C 5<<<<<<5 . 6 Q <!O;!21%%1J4!;I';! " G'H ! -6% !'*-!
) ! 6% <!315!J.!,'*!J();.!!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! ! 5" - 5 C - <<<<<< - 5 % ! 7" 5 - 7C 5 <<<<<< - 7% 5 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GSH!
'*-!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! ! ." - . C - <<<<< - . ) ! +" . - +C . - <<<<< - +) . 5!!!!!!!!!!!!!!!!!!!!!!!!!!GVH!
!
21(!4$);'8%.!.%./.*;4! 7" 5 7C 5<<<<<5 7% !'*-! +" 5 +C 5<<<<<5 +) !2(1/!!<!
\.;! . !8.!'!L()/);):.! '%3 !(11;!12!$*);0!)*!41/.!.M;.*4)1*!2).%-!12! !) G& H <!6I.*!;I.!
L1%0*1/)'%4!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 9 G $ H G +H ! + G + & . 8 H 5!!! " * $ * % 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GWH!
8$5$
!
!
S
'(.!L1%0*1/)'%4!)*! !) G& HZ +[ !12!-.&(..!-6<!+,;$'%%05!.',I 9 G $ H G +H !)4!;I.!L(1-$,;!12!6,
)((.-$,)8%.!L1%0*1/)'%4! 2$ G +H !1:.(! !) G&H !12!-.&(..!-1!^.!,1*4)-.(!'%%!;I.4.!
L1%0*1/)'%4!'4!(.L(.4.*;';):.4!!12!;I.!.%./.*;4!12!;I.!()*&!!12!L1%0*1/)'%4!/1-! + ' & " !
1:.(!;I.!2).%-! !) G& H 5!JI),I!)*!;I)4!F.L1(;!4I'%%!8.!-.*1;.-!80! "'& <!
!
6I.!L1%0*1/)'%4! 9 G$ H G +H !&.*.(';.!)* "'& !)-.'%4!!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ('$ 5& 5% ]! G 9 G$ H G +HH 5!!!! " * $ * % 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GUH!
!
,'%%.-!!9:':-;8$<:=,-:6$=>:,?@=:6!G!"#?@=:6H,J);I!L'('/.;.(4!'5!&!'*-!%<!6I.4.!,1-.4!'(.!
'%%!.N$):'%.*;!21(!2)M.-!:'%$.4!12!'5!&!'*-!%1,`$(;I.(/1(.5!);!)4!L(1:.*!)*!Z"5!L(112!12!
6I.1(./!"<"[!;I';!)2! 0 ' !)4!,0,%),5!);4!4$8&(1$L!5!12!)*-.M!%!,1*4)4;4!12!;I.!%#L1J.(4!G%#
(.4)-$.4H!/1-!'<!+!J.%%c?*1J*!L(1L.(;0!)4!;I';! 0 ' !)4!,0,%),!)2!'*-!1*%0!)2!
' $ PC5 S5 * # 5 C * # ! # / "Q 5!JI.(.!*!)4!'*!1--!L()/.!'*-! # !'*0!)*;.&.(! / " <!d.*,.5!J.!
I':.!;I';!21(!;I.4.!4L.,)'%!:'%$.4!12!'5!;I.!&(1$L!5!'4!-.2)*.-!'81:.!'4!'!4$8&(1$L!12! 0 ' !
J);I!)*-.M!%5!)4!)-.*;),'%!;1! 0 ' % 5!;I.!4$8&(1$L!12!'%%!%#(.4)-$.4!Z"5!6I.1(./!C<"[<!
!
^.!-.2)*.!'!4L.,)'%!4$8,%'44!12!!"#,1-.4!)*!;I.!21%%1J)*&!J'0<!
!
Definition 1
AB,%3:,6>C9-@>*,5,@B, 0 ' ,D$%3,$'=:+,%,$6,$=:'%$?;8,%@,%3:,6>C9-@>*, 0 ' E ,@B,,E#*@D:-6,GE#
-:6$=>:6,E@=,'H,@B,%3:,:8:E:'%6,@B, 0 ' ,B@-,6@E:,E/,E! " G'H /,%3:',%3:,?@=:, ('$ 5& 5% ,$6!?;88:=,
;',E#-:6$=>:,?@=:/,$B,E,$6,%3:,6E;88:6%,F;8>:,6;%$6B7$'9,%3$6,$=:'%$%71,
,
`(1/!;I.!(./'(?4!'81:.5!);!J)%%!8.!,%.'(!;I';!21(! ' $ PC5 S5 * # 5 C * # ! # / "Q !;I.!!"#,1-.4!!
'(.!%#(.4)-$.!,1-.45!21(!'%%!(.%.:'*;!:'%$.4!12!&5!%!'*-!$1,
O*!&.*.('%5!;I.!)*-.M!! A '5E !12!'!4$8&(1$L! 0 ' E !J);I!(.4L.,;!;1! 0 ' !J)%%!*1;!8.!.N$'%!;1!E1!
O*!4;.'-!J.!I':.!!;I.!21%%1J)*&!(.%';)1*4<!O2!J.!J();.!21(!'*!'(8);('(0!'#:'%$.!!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ' ! C#" *C #C <<<<<< *8 #8 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GTH!
!
J);I!1--!L()/.4! *C 5<<<<<<5 *8 5!;I.*!);!21%%1J4!2(1/!Z"5!6I.1(./!C<C[!;I';!
!
!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!! GE5 " GC#" HHGE5 " G *C #C HH<<<<<<<<<<GE5 " G *8 #8 HH 5!!!!!! #" * C !! 0 !!E,1--5!!!!!!!!!!!!!!!!GXH!
!
{
A ' 5E !_!!
!
!
!!!!!!!!!!!!!!!!!!!!!!! CGE5 C#" & C HGE5 " G *C#C HH<<<<<<<<<<<<G E5 " G *8 #8 HH 5!!!!!!!! #" % C !! 1 !!E,!.:.*<!!!!!!!G"YH!
!
V
!
315!J.!,'*!)//.-)';.%0!4;';.!;I.!21%%1J)*&!;I.1(./<!
!
Theorem 2
AB, ('$ 5& 5% /, " * $ * % /,$6,;',E#-:6$=>:,?@=:,@B,=$E:'6$@', ' & " G'H a % /,C;6:=,@',6@E:,6>C9-@>*!
0 ' E 2 0 ' /,%3:'! % ! A ' 5E <!
!
6I.!*.M;!.M'/L%.!G,2<!'%41!Z!!5KM'/L%.!C<S[H!4I1J4!;I';!%,,)4!*1;!'%J'04!.N$'%!;1!E1!
!
Example 3
`1(! ' ! SV !J.!2)*-! " GSVH ! " GVH" GXH _!S<W!_!CS!'*-!
!
0 SV ! P"5 C5 S5 U5T5""5"R5"S5"W5"U5"X5 CC5 &"5 &C5 &S5 &U5 &T5 &""5 &"R5 &"S5 &"W5 &"U5 &"X5 &CCQ
!
`)(4;!J.!;'?.! 5 ! 0 SV C _!P"5 S5"W5"X5 &"S5 &""Q !GE,G!CH!JI),I!I'4!!)*-.M! A SV5C ! !
GC5 " GVHHGC5 " GXHH ! GC5 SHGC5 WH ! S <!3)*,.! . ]!( & % !I'4!;1!8.!'!4$8&(1$L!12!55!J.!,'*!
;'?.!21(!)*4;'*,.! & ! S !1(! & ! "W <!+,;$'%%05!1*.!,1$%-!;'?.!'*0!L1J.(!12!S5!4)*,.!'%%!Sc
L1J.(4!/1-!SV!'(.!)*!51!\.;! & ! S< !6I.*!-HG!1(- SV GSH ! W 5! . ! 5 !'*-!
$
) ! " GSVH a W ! S _!6%1!315! 6 ! " !'*-! % ! S !e*.!18;')*4!21$(!G%GSH!Cc(.4)-$.!,1-.4! (SV5S5S
5!
" * $ * S 5!1:.(! !) GSH !12!-)/.*4)1*! SV & CS a S ! RX <!^I.*!;'?)*&! & ! "W !;I.!-';'!
8.,1/.! - ! 1(- SV G"WH ! R 5! . ! P"5"W5 &"SQ 5! " GSVH a R ! T 5! 6 ! C 5! % ! S <!!O*!;I)4!,'4.!1*.!
$
J)%%!18;')*!21$(!Cc(.4)-$.!,1-.4! (SV5"W5S
5! " * $ * S 5!12!%.*&;I!SV!1:.(! !) G"WH !'&')*!12!
-)/.*4)1*!RX<!
b.M;!J.!;'?.! 5 ! 0 SV S !_!P"5"W5 &"SQ !GE,G,SH!!J);I!)*-.M! A SV5S ! GS5 SHGS5 WH ! T <!b1J!1*.!
,1$%-!;'?.!'*0!L1J.(!12!"W!!21(!&1!71*4)-.(! & ! "W 5!&):)*&! - ]! 1(- SV G"WH ! R 5! . ! 5 !'*-!
$
) ! " GSVH a R ! T ! 6% <!315! 6 ! " 5! % ! T !'*-!1*.!18;')*4!.)&I;!Sc(.4)-$.!,1-.4! (SV5"W5T
5!
" * $ * T 5!1:.(! !) G"WH !12!-)/.*4)1*! SV & CS a T ! SC <!3)/)%'(%05!1*.!18;')*4!.)&I;!Tc
$
(.4)-$.!,1-.4! (SV5"W5T
!1:.(! !) G"WH <!6I.!%';;.(!,'4.!!GE,G!TH!-./1*4;(';.4!;I';!%,,'*!8.!
.N$'%!;1!E,JI.*! ' 3 PC5 S5 * # 5 C * # Q <!O*!;I.!2)(4;!;J1!,'4.4!1*.!I'4 % % E < !!!!!!!!!!!!!!!!!!!!!!"!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!
!
+*1;I.(!!*1*c;():)'%!.M'/L%.!12!'!&(1$L!!5!J);I! . 2 5 2 0 ' !5! 5 ! 0 ' E 5!4';)420)*&!
A ' 5 E ! E 5!JI)%.! ' 3 PC5 S5 * # 5 C * # Q !)4!L(.4.*;.-!8.%1J!G,2<!'%41!Z"5!KM'/L%.!C<V[H<!
!
Example 4
6'?.! ' ! RW !'*-! & ! "U <!3)*,.!1(- RW G"UH !_!C5!;I.!&(1$L!.!_!P"5!"UQ<!
b1J5!J.!-.2)*.! 5 ]! 0 RWR ! !P"5!"U5!! &" 5! &"U Q<!d.*,.5!! 0 RWR !!_!S!_!
" GRWH aGR5 CHGR5 WH ! "C a R <5!'*-!;I.!)*-.M!12!5!)4!R<!
!
W
e*!;I.!1;I.(!I'*-5!21(! E ! W !J.!2)*-! A RW5W ! GW5 CHGW5 WH ! "C 4 W <!!!!!!!
^.!,1*,%$-.!;I';!;I.!,I1),.! 5 ! 0 RWR !-.%):.(4!;I(..!.N$):'%.*;!Rc(.4)-$.!,1-.4!1:.(!
!) G"UH <!6I.4.!,1-.4!'(.!&.*.(';.-!80!L1%0*1/)'%4! 9 G$ H G +H 5! " * $ * R 5!12!-.&(..!S<!K',I!
4$,I!L1%0*1/)'%!)4!;I.!L(1-$,;!12!;J1!)((.-$,)8%.!L1%0*1/)'%4!1:.(! !) G"UH !12!-.&(..!C<!!
+!,I1),.! 5 ! 0 RW W ! ! P"Q !)4!*1;!L144)8%.5!4)*,.! . J1$%-!*1;!8.!,1*;')*.-!)*!5<!!!!!!!!!!!!!!!! !!!!!
6I.!*.M;!,'4.!J.!4I'%%!.ML%1(.!)4! 5 ! 0 SV _! P"5T5"U5"X5&"5&T5&"U5&"XQ !JI),I!I'4!)*-.M!
% ! A SV5R ! GR5 " GVHHGR5 " GXHH ! GR5SHGR5WH ! R <!O2!J.!;'?.! & ! T 5!;I.!&(1$L! . ! P"5T5"X5"UQ !
R
I'4!)*-.M!!C!J);I!(.4L.,;!;1!55!'*-!41! 6 ! C <!`$(;I.(/1(.5! - ! S !'*-! ) ! CS a S ! W !)*!
$
',,1(-'*,.!J);I! 6% ! W5 ! 6 ! C !'*-! % ! R <!d.*,.5!;I.(.!'(.!;I(..!Rc(.4)-$.!,1-.4! ( SV
5T 5 R 5!
" * $ * R 5!12!%.*&;I!SV!1:.(! !) GTH <!K',I!12!;I.!/)*)/'%!&.*.(';1(4!)4!;I.!L(1-$,;!12!C!G_6H!
)((.-$,)8%.!L1%0*1/)'%4!12!-.&(..!S!1:.(! !) GTH <!O*!;I)4!,'4.!J.!'%41!I':.! % ! EG! RH <!
b1J5!%.;! 5 ! 0 SV ! P"5"XQ !J);I!)*-.M! % ! A SV5W ! GW5SHGGW5WH ! "C <!6I.!1*%0!L144)8%.!*1*c
W
;():)'%!&(1$L! . , 5 !)4! . ! 5 ! P"5"XQ !&.*.(';.-!80! & ! "X <!6I.!,1((.4L1*-)*&!
L'('/.;.(!:'%$.4!'(.!-,]_!1(- SV G"XH ! C 5! " ! "C 5! 6 ! " 5! % ! "C <!315!;I.(.!'(.!;J.%:.!Wc
(.4)-$.!,1-.4!12!%.*&;I!SV!1:.(! !) G"XH !12!-)/.*4)1*! ' & " G'H a % ! SV & CS a "C ! SR <!
O2!1*.!-.2)*.4! 5 ! P"5T5"U5"XQ !;I.(.!)4!*1!E#:'%$.!4$,I!;I';! 5 ! 0 SV 5!'*-!;I.(.21(.!!"#
,1-.4!8'4.-!1*!;I)4!&(1$L!5!-1!*1;!8.%1*&!;1!;I.!2'/)%0!12!E#(.4)-$.!,1-.4<!f144)8%.!
:'%$.4!21(!&,,'(.!!T5!"U5!"X!1(!L1J.(4!12!;I.4.!:'%$.4<!6'?)*&! & ! T !&):.4! - ! S 5! ) ! W 5!
$
% ! W 5! 6 ! " <!315!;I.(.!'(.!4)M!!"#,1-.4! ( SV
5T 5 W 5! " * $ * W 5!12!%.*&;I!SV!1:.(! !) GTH< !!!!!!!#!!!
E
!
!!
`1(!/1(.!.M'/L%.4!12!!"#,1-.4!'*-!21(!;I.)(!(.%';)1*4I)L!J);I!4)/)%'(!,1-.4!'%(.'-0!
L(.4.*;!)*!;I.!%);.(';$.5!J.!(.2.(!;1!Z"[<!O*!;I)4!F.L1(;!J.!4I'%%!21,$4!1*!;I.!)-./L1;.*;!
&.*.(';1(4!21(!!"#,1-.45!.4L.,)'%%0!21(!'!,1$L%.!12!2'/)%).4!12!%#(.4)-$.!,1-.4<!+2;.(!'!
&.*.('%!)*;(1-$,;)1*!;1!)-./L1;.*;!&.*.(';1(4!21(!,0,%),!,1-.4!)*!3.,;)1*!C5!J.!-.:.%1L!'!
*.J!'LL(1',I!;1!;I)4!;1L),!)*!3.,;)1*!R!80!-.2)*)*&!'!/';()M!I,,;I.!,1%$/*4!12!JI),I!
0).%-!;I.!L()/);):.!)-./L1;.*;!&.*.(';1(4!21(!,0,%),!,1-.4!12!%.*&;I!'1!O*!L'(;),$%'(5!J.!
'LL%0!;I)4!'LL(1',I!;1!!"#,1-.4!)*!3.,;)1*!S<!O*!3.,;)1*4!V5!W5!U!'*-!T5!J.!4;$-0!;I.!
)-./L1;.*;!&.*.(';1(4!21(!!"#,1-.4!12!%.*&;I! * 5!! C * 5! * # !'*-! C * # 5!(.4L.,;):.%0<!+%%!
;I.4.!,1-.4!8.%1*&!;1!;I.!4$82'/)%0!12!%#(.4)-$.!,1-.4<!6I.!(.'41*!;1!;(.';!;I.!,'4.4!
' ! * !'*-! ' ! * # !4.L'(';.%05!)4!;I';!;I.!,'4.! ' ! * !'*-!&!'*!'(8);('(0!L()/.!L1J.(!J);I!
G *5 & H ! " 5!)4!-.'%;!J);I!)*!;I.!%);.(';$(.!'4!'!&.*.('%)E';)1*!12!N$'-(';),!(.4)-$.!,1-.4!
GJI.*!&!)4!'!N$'-(';),!(.4)-$.!G,2<!Z"[HH5!JI)%.!;I.!,'4.! ' ! * # !)4!'!2$(;I.(!&.*.('%)E';)1*!
G,2<!Z"5S![H<!6I.!1;I.(!;J1!2'/)%).4!12!%#(.4)-$.!,1-.45!)<.<!;I.!,'4.4! ' ! C * !'*-! ' ! C * # 5!
'(.!%.44!?*1J*<!O*!1$(!*.M;!(.L1(;!J.!4I'%%!L(.4.*;!'!,1$L%.!12!(.2.(.*,.4!JI.(.!;I.4.!
,1-.4!'(.!)*;(1-$,.-<!`)*'%%05!)*!3.,;)1*!X5!J.!)*:.4;)&';.!'!!"#,1-.!JI),I!)4!*1;!
,1:.(.-!80!1*.!;I.!L(.:)1$4!,'4.45!80!;'?)*&! ' ! C" <!
!
U
^.!J1$%-!%)?.!;1!(./'(?!;I';!)*!L(',;),.5!;I.!I#/';()M!'LL(1',I!)4!*1;!'!:.(0!.22),).*;!
4$84;);$;.!21(!;I.!/.;I1-4!JI),I!J.(.!8().2%0!-)4,$44.-!)*!3.,;)1*!R5!4)*,.!1*.!*..-4!;I.!
,1.22),).*;4!12!;I.!1*.c8$;!I)&I.4;!L1J.(!12!+!)*!;88!)((.-$,)8%.!2',;1(4!12! + ' & " <!
!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
2. Idempotent generators of cyclic codes
\.;! 9 G +H !8.!;I.!/)*)/'%!&.*.(';1(!L1%0*1/)'%!12!'!,0,%),!,1-.!(!12!%.*&;I!'!1:.(!;I.!
2).%-! !) G&H 5!'*-!%.;! 3G +H 8.!);4!,I.,?!L1%0*1/)'%<!31!J.!I':.!!!!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! + ' & " ! 9 G +H3G +H 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!G""H!!!!!!!!!!!!!!!!!!!!!!
'*-!I.*,.5!)*! "'& !!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 9 G +H3G +H ! Y <!!!!!!!!!!!!<!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!G"CH!
!
Definition 5
J:%,(,C:,;,?7?8$?,?@=:1,K,*@87'@E$;8, :G +H $ ( /,@B,=:9-::,8:66,%3;','/,D3$?3,$6,;',$=:'%$%7,
:8:E:'%,@B,(,,$6,?;88:=,%3:,$=:E*@%:'%,9:':-;%@-,@B,,(1,
,
e*.!,'*!.'4)%0!-.():.!2(1/!;I)4!-.2)*);)1*5!;I';!.:.(0!,0,%),!,1-.!I'4!'*!)-./L1;.*;!
&.*.(';1(!JI),I!)4!$*)N$.!'*-!JI),I!4';)42).4!
,
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, :G +H C ! :G +H 1,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,G"RH!
,
+4!21(!;I.!$*)N$.*.44!12!4$,I!'*!)-./L1;.*;5!!J.!(.2.(!;1!;I.!;.M;811?45!.<&<!ZR5V[<!O;!)4!
'%41!J.%%c?*1J*5!;I';! :G +H !,'*!8.!18;')*.-!2(1/!;I.!(.%';)1*!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ;G +H 9 G +H 5 CG +H3G +H ! " 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!G"SH!
!
JI),I!I1%-4!21(!,.(;')*!L1%0*1/)'%4! ;G +H !'*-! CG +H !)*! "'& 5!8.,'$4.! 9 G +H !'*-! 3G +H !I':.!
*1!,1//1*!-):)41(45!.M,.L;!;I.!,1*4;'*;!L1%0*1/)'%!"<!#0!-.2)*)*&!
!
:G + H ! ; G + H 9 G + H
G"VH
'*-!80!'LL%0)*&!G"C!H!'*-!G"S!H!J.!2)*-!;I';!;I.!L1%0*1/)'%! :G +H )*!G"VH!4';)42).4!G"RH<!+!
2.J!J.%%c?*1J*!L(1L.(;).4!12!)-./L1;.*;!&.*.(';1(4!'(.!%)4;.-!)*!;I.!*.M;!;I.1(./<!
!
Theorem 6
G)H!!K',$=:E*@%:'%,*@87'@E$;8,$',(,D3$?3,$6,;,>'$%,B@-,(/,$6,;',$=:E*@%:'%,9:':-;%@-,@B,(g!
G))H!":8;%$@'!G"RH!$6,:&>$F;8:'%,%@, :G. $ H ! Y ,$B, 9 G. $ H ! Y ,;'=, :G. $ H ! " ,$B! 3G. $ H ! Y 5!
D3:-:,%3:, . $ /, " * $ * ' /,,;-:,%3:,' %3 ,-@@%6,@B,>'$%7,87$'9,$',6@E:,:+%:'6$@',B$:8=,@B! !) G&H g!
G)))H! ?G +H $ ( ,$B,;'=,@'87,$B, :G +H?G +H ! ?G +H 1!
!
!
T
`1(!;I.!L(1124!J.!'&')*!(.2.(!!;1!ZR5V[<!
!
e2!,1$(4.!;I.!L1%0*1/)'%! ; G +H 5!'*-!I.*,.! :G +H 5!,'*!8.!18;')*.-!80!,'((0)*&!1$;!
K$,%)-h4!'%&1();I/!;1!-.;.(/)*.!;I.!&,-!12!;J1!L1%0*1/)'%4<!6I1$&I!;I)4!'%&1();I/!)4!
(';I.(!.22),).*;5!)*!L(',;),.!1*.!!8.;;.(!,'*!$4.!41/.;)/.4!;I.!21%%1J)*&!L(1L.(;05!JI),I!)4!
2'4;.(!)*!&.*.('%<!!
!
!
Theorem 7
L3:,$=:E*@%:'%,9:':-;%@-, :G +H /,=:B$':=,$',G"VH5!!?;',C:,D-$%%:',;6,:8:E:'%,@B, "'& ,;6,
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! :G +H ! ' &" +3 iG +H 9 G +H 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!G"WH!
!
D3:-:, ' &" ,$6,%3:,$'F:-6:,@B,',$',%3:,B$:8=, !) G&H ,;'=,D3:-:! 3 iG +H ,6%;'=6,B@-,%3:,B@-E;8,
=:-$F;%$F:,@B,%3:,*@87'@E$;8, 3G +H <!
Proof
`)(4;!J.!(./'(?!;I';!;I.!-.&(..!12!;I.!L1%0*1/)'%!)*!;I.!(I4!12!G"WH!)4!%.44!;I'*!'/,4)*,.!
J.!)*;.(L(.;!;I';!L1%0*1/)'%!'4!'*!.%./.*;!12! "'& <!6I)4!)/L%).4!;I';!J.!I':.!;1!4$8;(',;!'!
;.(/! + ' & " !)*!,'4.!;I';!;I.!-.&(..!12!;I.!L1%0*1/)'%!)4!*1;!'!/$%;)L%.!12!;I.!,I'(',;.()4;),!
12!;I.!2).%-! !) G&H <!,\.;! . $ !8.!'!E.(1!12! 9 G +H <!6I.*!);!21%%1J4!)//.-)';.%0!2(1/!G"WH!
;I';! :G. $ H ! Y <!b1J5!'44$/.!;I';! . $ !)4!'!E.(1!12! 3G +H <!`(1/!;I.!L1%0*1/)'%!)-.*;);0!G""H!
);!21%%1J4!80!;'?)*&!-.():';):.4!;I';!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 9 G +H3 iG +H 5 9 iG +H3G +H ! '+ ' &" 5!
!
JI.(.!;I.!,1.22),).*;!'!)4!;'?.*!/1-$%1!&5!'*-!I.*,.!,'*!8.!,1*4)-.(.-!'4!'*!.%./.*;!12!
!) G&H !G(././8.(!;I';! G'5 &H ! " H<!315!!
!
!!!!!!!!!!!!!!!!!!!!!!!!!! :G. $ H ! ' &". $ 9 G. $ H3 iG. $ H ! ' &". $ <'. $ G ' &"H ! . $' ! " <!
!
+,,1(-)*&!;1!6I.1(./!W!G))H5!'*-!8.,'$4.!12!;I.!$*)N$.*.44!12!;I.!)-./L1;.*;!&.*.(';1(5!
);!21%%1J4!;I';! :G +H !.N$'%4!;I.!.ML(.44)1*!)*!;I.!(I4!12!G"WH<!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#!
!
!
Corollary 8
L3:,$=:E*@%:'%,9:':-;%@-,@B,;,?7?8$?,?@=:,@B,8:'9%3,',@F:-,%3:,B$:8=, !) GCE H ,$6,:&>;8,%@,
,
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, :G +H ! 9 G +H3 @ G +H ! 9 G +H3 : G +H 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!G"UH!
!
D3:-:/, 3@ G +H ,;'=, 3: G +H ,;-:,%3:,*@87'@E$;86,@C%;$':=,B-@E, 3G +H ,C7,%;4$'9,@'87,%3:,@==,
-:6*:?%$F:87,%3:,:F:',*@D:-6,@B,+,$', 3G +H 1,
!
X
,
3)*,.! ' ! " !/1-!C5!;I.!L(112!)4!4;(')&I;21(J'(-<!`1(!;I.!8)*'(0!,'4.5!6I.1(./!U!'*-!);4!
71(1%%'(0!'(.!L(.4.*;.-!)*!ZV5!7I<!T5!f(18%./!G"UH[5!'4!'*!.M.(,)4.<!
!
O*!L'(;),$%'(5!J.!,'*!'LL%0!;I.!'81:.!.ML(.44)1*4!21(! :G +H ;1!&.*.('%)E.-!(.4)-$.!,1-.4!
JI),I!J.(.!)*;(1-$,.-!)*!Z"[!G,2<!3.,;)1*!"H<!^.!4I'%%!&):.!'!2.J!.M'/L%.4<!
!
Example 9
6'?.! ' ! U !'*-! & ! C <!3)*,.!1(- U GCH ! R 5!;I.!4L%);;)*&!2).%-!12! + U 5 " !1:.(! !) GCH !)4!
!) GC R H <!6I.!&.*.(';)*&!.%./.*;!12!;I.!/$%;)L%),';):.!&(1$L! !) GC R H j !)4! 6 !-.2)*.-!80!!
6 R 5 6 5 " ! Y <!d.(.5!J.!I':.!',,)-.*;%0! 6 ! . <!
6I.!2',;1()E';)1*!12! + U & " !)*;1! + & " !'*-!;J1!)((.-$,)8%.!L1%0*1/)'%4!12!-.&(..!R!1:.(!
!) GCH !)4!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! + U 5 " ! G + 5 "HG + R 5 + 5 "HG + R 5 + C 5 "H <!
!
O2!J.!-.2)*.! 9 G +H ]! + R 5 + 5 " 5!;I.*! 3G +H ! + S 5 + C 5 + 5 " <!!71(1%%'(0!T!*1J!L(1:)-.4!$4!
J);I! :G +H ! G + R 5 + 5 "H + ! + S 5 + C 5 + <!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !
!
Example 10
6'?.! ' ! V !'*-! & ! "X <!^.!J();.! + V & " ! G + & "H' V G +H <!3)*,.!1(- V G"XH ! C 5!!;I.!
2',;1()E';)1*!12! ' V G +H !)*! !) G"XHZ +[ !,1*;')*4! S a C ! C !)((.-$,)8%.!L1%0*1/)'%4!12!
-.&(..!C<!KML%),);%05!J.!2)*-!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!! + V & " ! G + & "HG + C & S + 5 "HG + C 5 V + 5 "H <!
!
\.;! 9 G +H ! G + & "HG + C 5 V + 5 "H 5!;I.*! 3G +H ! + C & S + 5 " <!+,;$'%%05!;I)4!&.*.(';)*&!
L1%0*1/)'%!-.2)*.4!'*!.M;.*-.-!!"#,1-.<!6I.1(./!U!&):.4!;I.!)-./L1;.*;!
!
!!!!!!!!!!!!!!!!!!!! :G +H ! V&" +9 G +HGC + & SH ! SGS + S & V +R & V + C 5 S + 5 CH <!
!
O*-..-5!1*.!,'*!:.()20!;I';!)*! "V"X !1*.!I'4! :G +H C ! :G +H 5!)2!;I.!,1.22),).*;4!'(.!,1*4)-.(.-!
;1!8.!.%./.*;4!12! !) G"XH< !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! !
!
Example 11
6'?.! ' ! W !'*-! & ! U <!^.!J();.!!
!
!!!!!!!!!!!!!!!! + W & " ! G + & "H' C G +H' R G +H' W G +H ! G + & "HG + 5 "HG + C 5 + 5 "HG + C & + 5 "H !
!!!!!!!!!!!!!!!!!!!!!!!!!!!_! G + & "HG + 5 "HG + & CHG + & SHG + 5 CHG + 5 SH <!
!
^.!-.2)*.! 9 G +H ! G + 5 "HG + C 5 + 5 "H ! +R 5 C + C 5 C + 5 "H 5!&):)*&!()4.!;1!!
!
"Y
3G +H ! G + & "HG + C & + 5 "H ! +R & C + C 5 C + & " <!3;(),;%0!4L.'?)*&5!;I)4!L1%0*1/)'%! 9 G +H !
-1.4!*1;!&.*.(';.!'!!"#,1-.<!6I.1(./!U!&):.4!21(!;I.!)-./L1;.*;!,1((.4L1*-)*&!;1! 9 G +H !
;I.!.ML(.44)1*!
!
!!!!! :G +H ! W&" +9 G +H3 iG +H ! W&" +G +R 5 C + C 5 C + 5 "HGR + C & S + 5 CH _! V + V 5 + R 5 V + 5 S <!
!
+&')*5!1*.!,'*!.'4)%0!:.()20!;I.!)-./L1;.*,0!(.%';)1*! :G +H C ! :G +H !)*! "WU <!!!!!!!!!!!!!!!!!!!!!!"!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!
!
Example 12
6'?.! ' ! "" !'*-! & ! S <!3)*,.!1(- "" GSH _V5!;I.!L1%0*1/)'%! ' "" G +H !,'*!8.!2',;1()E.-!)*;1!
;J1!)((.-$,)8%.!L1%0*1/)'%4!1:.(! !) GSH !12!-.&(..!V<!O*!.ML%),);!21(/!J.!,'*!J();.!
!
!!!!!!!!!!! +"" 5 " ! G + 5 "HG + V 5 6+ S 5 + R 5 + C 5 6 C + 5 "HG + V 5 6 C + S 5 + R 5 + C 5 6+ 5 "H 5!
!
JI.(.! 6 !4';)42).4! 6 C 5 6 5 " ! Y <!+,;$'%%05!'!L()/);):.!"" %3 !(11;!12!$*);0!)4!-.2)*.-!'4!'!
E.(1!12!1*.!12!;I.4.!;J1!V %3 !-.&(..!L1%0*1/)'%4<!O2!J.!-.2)*.!
9 G +H ! + V 5 6+ S 5 + R 5 + C 5 6 C + 5 " 5!;I.*! 3G +H ! + W 5 6+ V 5 6+ S 5 6 C + C 5 6 C + 5 " 5!'*-!
41! 3 @ G +H ! 6+ V 5 6 C + <!!
71*4.N$.*;%05!$4)*&!71(1%%'(0!T5!J.!2)*-!!
!
:G +H ! 3 @ G +H 9 G +H ! 6+ "Y 5 6 C + X 5 6+ T 5 6+ U 5 6+ W 5 6 C + V 5 6 C + S 5 6 C + R 5 6+ C 5 6 C + <!
!
#0!'!4;(')&I;21(J'(-!,1/L$;';)1*!J.!:.()2).-!;I';!;I)4!.ML(.44)1*!4';)42).4!;I.!
)-./L1;.*,0!(.%';)1*!)*! """S <!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!
!
!
!
Remark 13
#.,'$4.!12!;I.!$*)N$.*.44!12!;I.!)-./L1;.*;!&.*.(';1(5!J.!I':.!'4!'!,1*4.N$.*,.!12!;I.!
(.%';)1*4!G"VH!'*-!G"WH!!;I.!.N$'%);).4! ; G +H ! ' &" +3i G + H !'*-! CG +H ! ' &" +9 i G +H <!6I.(.21(.5!
)*!4;.'-!12!G"SH!J.!I':.!)*! "'& !;I.!)-.*;);0!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ' &" +G 9 G +H3i G +H 5 9 i G +H3G +HH ! " 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!G"TH!
!
1(!.N$):'%.*;%05!!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ' &" +G 9 G +H3i G +H 5 9 i G +H3G +HH ! " !/1-! + ' & " 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!G"XH!!
!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!JI),I!12!,1$(4.!)4!18:)1$4%0!;($.<!
!
!
""
!
!
6I.(.!)4!4;)%%!'*1;I.(!;11%!;1!,1/L$;.!)-./L1;.*;!&.*.(';1(4!12!,0,%),!,1-.4<!6I)4!;11%!)4!
8'4.-!1*!;I.!41c,'%%.-!$'F:-6$@',->8:!G,2<!.<&<!ZV5!7I<!U[H<!
!
Theorem 14!
J:%, ?G +H ! ? ' &" + ' &" 5 ? ' &C + ' & C 5 <<<<<<< 5 ?Y ,C:,;,*@87'@E$;8,$', "'& 1,L3:',%3:,?@:BB$?$:'%6, ?$ /,
' &"
Y * $ * ' & " /,?;',C:,@C%;$':=,C7, ?$ ! ' &" 7 ?G. M H. &$M /,D3:-:, . ,$6,;,*-$E$%$F:, ' %3 ,-@@%,@B,
M !Y
>'$%7,$',6@E:,:+%:'6$@',B$:8=,@B,, !) G&H 1!
!!
+LL%0)*&!;I)4!L(1L.(;0!;1!;I.!)-./L1;.*;!&.*.(';1(! :G +H !12!'!,0,%),!,1-.!0).%-4!;I.!
21%%1J)*&!;I.1(./<!
!
Theorem 15
J:%, :G +H ! :' &" + ' &" 5 :' & C + ' & C 5 <<<<<<<<<< 5 :Y ,C:,%3:,$=:E*@%:'%,9:':-;%@-,@B,;,?7?8$?,?@=:,(,/,
9:':-;%:=,C7,%3:,E$'$E;8,*@87'@E$;8, 9 G +H 1,L3:',%3:,?@:BB$?$:'%6, :$ /, " * $ * ' & " /,?;',C:,
@C%;$':=,C7, :$ ! ' &" 7 . & $M /,D3:-:,N,$6,%3:,6:%,@B,:+*@':'%6,,@B,%3:,'@'<:-@6, . M ,@B, 9 G +H 1!
M$N
!
Example 16
^.!4I'%%!'LL%0!6I.1(./!"V!!;1!;I.!!"#,1-.!J);I!L'('/.;.(4! ' ! "R 5! & ! R !'*-! % ! S 5!
&.*.(';.-!80!;I.!L1%0*1/)'%! 9 G"H G +H ! + R & + & " <!6I)4!,1-.! ("R" 5R5 S 5!!)4!'%41!4;$-).-!)*!Z"5!
KM'/L%.4!C<C5!R<"!'*-!S<R[<!6I.!E.(14!12! 9 G"H G +H !'(.! . 5! . R !'*-!! . X <!d.*,.5!);!21%%1J4!
2(1/!6I.1(./!"V!;I';!
"C
!! :$ ! "R &" 7 :G. M H. &$M _! " 5 . &C$ 5 . &S$ 5 . &V$ 5 . &W$ k . &U $ 5 . T$ 5 . &"Y$ 5 . &""$ 5 . &"C$ <!
M !Y
6I.!L()/);):.!"R %3 !(11;!12!$*);0! . !)4!-.2)*.-!80! 9 G"H G. H ! Y 5!JI),I!L(1:)-.4!$4!J);I!;I.!
)-.*;);).4]! . R ! . 5 " 5! . S ! . C 5 . 5! . V ! . C 5 . 5 " 5! . W ! . C & . 5 " 5!
. U ! &. C & . 5 " 5! . T ! &. C & " 5! . X ! . & " 5! . "Y ! . C & . 5! . "" ! &. C 5 . 5 " !'*-!
"C
. "C ! ! . C & " <!#0!'%41!/'?)*&!$4.!12!;I.!)-.*;);0! 7 . M ! Y 5!);!)4!.'40!;1!-.():.!
M !Y
:" ! :R ! :X ! " 5! :C ! :W ! :V ! &" 5! :U ! :T ! :"" ! " !'*-! :S ! :"C ! :"Y ! Y <!
d.*,.5!;I.!)-./L1;.*;!&.*.(';1(!12!;I.!,1-.!-.2)*.-!80! 9 G"H G + H !)4!.N$'%!;1!
!
!!!!!!!!!!!!!!!!!!!!!! : G"H G +H ! + "" 5 + X 5 + T 5 + U & + W & + V 5 + R & + C 5 + 5 " <!
!
61!'LL%0!6I.1(./!U5!J.!2)(4;!-.;.(/)*.!;I.!L1%0*1/)'%! 3G +H ]! +"R & " a 9 G + H <!^.!2)*-!
!!
!!!!!!!!!!!!!!!!!!!!!!!!!!! 3G +H ! +"Y 5 + T 5 + U 5 + W & + V & + S 5 + C & + 5 " !
!
"C
!'*-!*.M;!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 3i G +H ! + X & + U 5 + W 5 + S & + R & + & " <!!
!
^I.*!.:'%$';)*&! : G"H G +H ]! 9 G +H3i G +H 5!J.!.*-!$L!J);I!;I.!4'/.!.ML(.44)1*!JI),I!J'4!
-.():.-!8.21(.<!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!
3)/)%'(%05!J.!,'*!-.;.(/)*.!;I.!)-./L1;.*;!&.*.(';1(!12!;I.!,1-.! ("RR 5R5 S !-.2)*.-!80!);4!
/)*)/'%!L1%0*1/)'%! 9 G RH G +H ! + R 5 + C & " 5!J);I!E.(14! . S 5! . "C !'*-! . "Y <!6I.!(.4$%;!)4!!
!
!!!!!!!!!!!!!!!!!!!!! : GRH G +H ! +"C & +"" 5 +"Y & + T & + U 5 + W 5 + V 5 + S 5 + C <!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#!!!!!!!!!!!!!!!!!!!!!!!!
! !
!
!
Example 17
`1(! ' ! "V 5!J.!I':.! 0"V ! P"5 C5 S5 U5T5""5"R5"SQ !'*-! " G"VH ! T <!6I.!,0,%1;1/),!
L1%0*1/)'%! '"V G +H !,'*!8.!18;')*.-!2(1/!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! +"V & " ! G + & "H' R G +H' V G +H'"V G +H 5!
&):)*&!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! '"V G +H ! +T & + U 5 +V & + S 5 +R & + 5 " <!
!
^.!,1*4)-.(! '"V G +H !'4!'!L1%0*1/)'%!1:.(! !) GCH !'*-!J();.!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! '"V G +H ! +T 5 + U 5 +V 5 + S 5 + R 5 + 5 " <!
!
3)*,.!1(- "V GCH ! S 5!J.!,'*!2',;1()E.! '"V G +H !)*! ""VC !)*;1!;J1!)((.-$,)8%.!L1%0*1/)'%4!12!
-.&(..!S!G,2<!Z"53.,;)1*"[H!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! '"V G +H ! G + S 5 + 5 "HG + S 5 +R 5 "H <!
!
6I.!,1/L%.;.!2',;1()E';)1*!12! +"V & " !)*! ""VC !)4!
!
!!!!!!!!!!!!! +"V & " ! G + 5 "HG + C 5 + 5 "HG + S 5 + R 5 + C 5 + 5 "HG + S 5 + 5 "HG + S 5 + R 5 "H <!
!
\.;! . !8.!'!L()/);):.!"V %3 !(11;!12!$*);05!-.2)*.-!'4!'!E.(1!12! + S 5 + 5 " <!6I.*!);!,'*!.'4)%0!
8.!-.():.-!;I';!'/1*&!;I.!)((.-$,)8%.!L1%0*1/)'%4!)*!;I.!'81:.!2',;1()E';)1*!1*.!I'4!;I.!
21%%1J)*&!-)4;()8$;)1*!12!E.(14]!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! + S 5 + 5 " !!!!!!!!!!!!!!!!!!!J);I!E.(14!!! . " 5 . C 5 . S 5 . T 5!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! + S 5 + R 5 " !!!!!!!!!!!!!!!!!!J);I!E.(14!!! . R 5 . W 5 . "C 5 . X 5!
!
"R
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! + S 5 + R 5 + C 5 + 5 " !!!J);I!E.(14!!! . U 5 . "S 5 . "R 5 . "" 5!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! + C 5 + 5 " !!!!!!!!!!!!!!!!!!!J);I!E.(14!!! . V 5 . "Y !5!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! + 5 " !!!!!!!!!!!!!!!!!!!!!!!!!!J);I!E.(1!!!! . Y <!
`1(! & ! C 5!;I.!1*%0!!E#:'%$.4!%.44!;I'*!T!'*-!4$,I!;I';! C $ 0"V E !'(.!"5!R5!V!'*-!U5!JI)%.!
0 "V ! 0 "V ! 0 "V ! 0 "V ! 0 "V <!315!;I.!1*%0!8)*'(0!E#(.4)-$.!,1-.!)4!;I.!;():)'%!,1-.!
"
R
V
U
("V" 5 C5" !JI),I!)4!'!"c(.4)-$.!,1-.!'*-!JI),I!)4!18;')*.-!80!,I114)*&! 5 ]! 0 "V 8 . !( C % <!
^I.*!J.!,I114.! 5 ]! . ! P"5 C5 S5TQ 5!);!21%%1J4!;I';! % ! T a S ! C 5!'*-!J.!18;')*!;J1!
8)*'(0!.N$):'%.*;!!"#,1-.4! ("V$ 5 C5 C 5! " * $ * C 5!JI),I!'(.!*1;!E#(.4)-$.!,1-.4!21(!'*0!E1!!
!
b.M;5!J.!,1*4)-.(! '"V G +H !)*!;I.!L1%0*1/)'%!()*&! !) GSHZ +[ <!3)*,.!1(- "V GSH ! C 5!J.!*1J!
I':.!'!2',;1()E';)1*!)*;1!21$(!L1%0*1/)'%4!12!-.&(..!C!JI),I!'(.!)((.-$,)8%.!)*! !) GSHZ +[ !
!
!!!!!!!!!!!!!!!!!!!!! '"V G +H ! G + C 5 + 5 6 HG + C 5 + 5 6 C HG + C 5 6 + 5 6 HG + C 5 6 C + 5 6 C H 5!
!
JI.(.! 6 !4';)42).4! 6 C 5 6 5 " ! Y <!
^.!,1*,%$-.!;I';!JI.*!;'?)*&! & ! S $ 5 _!P"5SQ5!;I.(.!'(.!!! 0"V a 5 !_ " G"VH a C ! S !
$
5! " * $ * S <!3)*,.! 5 ! 0"V C 5!;I.4.!,1-.4!'(.!.:.*!Cc
N$';.(*'(0!.N$):'%.*;!!"#,1-.4! ("V5S5S
(.4)-$.!,1-.4!G1:.(! !) GSH H<!^.!-.2)*.5 9 G"H G +H ]! + R 5 + 5 6 5! 9 G RH G +H ]! + C 5 + 5 6 C 5!
9 G CH G +H ]! + C 5 6+ 5 6 !'*-! 9 G SH G +H ]! + C 5 6 C + 5 6 C <!O*!1(-.(!;1!18;')*!;I.!)-./L1;.*;!
&.*.(';1(!12!;I.!,1-.! ("V" 5 S5 S ]_ G 9 G"H G +HH 5!J.!2)(4;!-.;.(/)*.!;I.!,1((.4L1*-)*&!,I.,?!
L1%0*1/)'%!!
!
!!!!! 3 G"H G +H ! +"R 5 +"C 5 6 C +"" 5 +"Y 5 6+ T 5 6+ U 5 + W 5 6+ V 5 6 C + R 5 6 C + C 5 6+ 5 6 C <!
!
+LL%0)*&!6I.1(./!U!1(!71((1%%'(0!T!&):.4!;I.!)-./L1;.*;!&.*.(';1(!
!
! : G"H G +H ! +"S 5 + "R 5 6 C + "C 5 + "" 5 6+ X 5 6+ T 5 + U 5 6+ W 5 6 C + S 5 6 C + R 5 6+ C 5 6 C + 5 " <!
!
3)/)%'(%05!J.!18;')*!;I.!)-./L1;.*;!&.*.(';1(!12! G 9 G CH G +HH !80!4$,,.44):.%0!,1/L$;)*&!
!
!! 3 G CH G +H ! +"R 5 6+"C 5 +"" 5 +"Y 5 6+ T 5 6 C + U 5 6+ W 5 6+ V 5 6 C + R 5 + C 5 6 C + 5 6 C 5!
!
: G C H G +H ! 6+"S 5 6 C +"R 5 6+"C 5 6+"" 5 6 C + X 5 + T 5 6 C + U 5 6 C + W 5 + S 5 6+ R 5 + C 5 + <!
!
6I.!)-./L1;.*;!&.*.(';1(4! : G RH G +H !'*-! : G S H G +H !21%%1J!2(1/! : G"H G +H !'*-! : G C H G +H !80!
(.L%',)*&! 6 !J);I! 6 C 5!(.4L.,;):.%0<!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#!
!
!
!!
!
"S
!
!
!
!
Example 18
\.;! ' ! "Y !'*-!;'?.!2)(4;! & ! R <!6I.*! '"Y G +H ! + S & +R 5 + C & + 5 " <!3)*,.!1(- "Y GRH ! S 5!
) ! " G"YH a S ! " 5!'*-!41!;I.!L1%0*1/)'%! '"Y G +H !)4!)((.-$,)8%.!)*! !) GRHZ +[ <!6'?)*&!
"
5!&):.4!'!,I.,?!L1%0*1/)'%!
9 G +H ]! '"Y G +H !'4!&.*.(';1(!L1%0*1/)'%!12!;I.!!"#!,1-.! ("Y5R5"
3G +H ! + W 5 +V & + & " <!d.*,.5!;I.!)-./L1;.*;!&.*.(';1(!12!;I)4!,1-.!)4!!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!! :G +H ! "Y&" +3 iG +H 9 G +H ! & + X 5 +T & + U 5 + W 5 + V 5 + S & + R 5 + C & + <!
!
!
b.M;!J.!;'?.!! & ! X <!3)*,.! - ]! 1(- "Y GXH ! C !'*-! " G"YH ! S 5!J.!?*1J!;I';! '"Y G +H ,'*!8.!
J();;.*!'4!;I.!L(1-$,;!12!;J1!)((.-$,)8%.!L1%0*1/)'%4!1:.(! !) GXH 12!-.&(..!C<!`(1/!;I.!
.N$'%);0! +"Y & " ! G + & "H' C G +H' V G +H'"Y G +H 5!J.!-.():.!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! '"Y G +H ! G + C 5 6 + 5 "HG + C & G6 5 "H + 5 "H 5!
!
JI.(.! 6 !)4!-.2)*.-!!'4!1*.!;I.!(11;4!12!;I.!.N$';)1*! + C 5 + & " ! Y !G,2<!Z"5!KM'/L%.!U<"[H5!
JI)%.!;I.!1;I.(!(11;!)4! 6 R <!
^.!-.2)*.! 5 ]! . !( X % <!315! 5 ! P"5XQ !'*-! % ! " G"YH a C ! C <!d.*,.5!;I.(.!'(.!;J1!
$
5 " * $ * C 5!&.*.(';.-!80! 9 G"H G +H ! + C 5 6 + 5 " !'*-!
.N$):'%.*;!!Xc'(0!,1-.4!! ("Y5X5C
9 GCH G +H ! + C & G6 5 "H + 5 " 5!(.4L.,;):.%0<!^.!4I'%%!-.;.(/)*.!;I.!)-./L1;.*;!&.*.(';1(!12!
;I.!,1-.! G 9 G"H G +HH <!`)(4;5!J.!2)*-!21(!;I.!,I.,?!L1%0*1/)'%!!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 3 G"H G +H ! +T & 6 + U & 6 + W 5 +V & +R 5 6 + C 5 6 + & " <!
!
b.M;5!'LL%0)*&!6I.1(./!U5!J.!&.;!;I.!)-./L1;.*;!
!
!!!!! :G"H G +H !_! 6 +X 5 G6 5 "H +T & G6 5 "H + U & 6 + W & +V & 6 + S & G6 5 "H +R 5 G6 5 "H + C 5 6 + & " !
!!!!!!!!!!!!!!!!! ! 6+ X & 6 R + T 5 6 R + U & 6+ W & + V & 6+ S 5 6 R + R & 6 R + C 5 6+ & " <!
!
#0!'!4;(')&I;21(J'(-!,'%,$%';)1*!J.!:.()2).-!;I';! : G"H G +H C ! : G"H G +H <!
!
3)*,.! 6 R ! &6 & " 5!J.!,'*!18;')*!;I.!.ML(.44)1*!21(! 3 G CH G +H !80!(.L%',)*&! 6 !)*! 3 G"H G +H !
J);I! 6 R 5!'*-!4)/)%'(%05!80!;I.!4'/.!4$84;);$;)1*5!J.!,'*!&.;! : G C H G +H !2(1/! : G"H G +H <!6I.!
(.4$%;!)4!
!
!!!!!!! : G C H G +H ! 6 R + X & 6+ T 5 6+ U & 6 R + W & + V & 6 R + S 5 6+ R & 6+ C 5 6 R + & " <!
!
"V
!
^.!(./'(?!;I';!'!L()/);):.!"Y %3 !(11;! . !12!$*);0!,'*!8.!-.2)*.-!80!'4!'!E.(1!12! 9 G"H G + H !)*!!
!) GXC H ZM[<!6I)4!)/L%).4! . C ! &6. & " 5!1(!.N$):'%.*;%0! 6 ! &G. 5 . &" H <!!!!!!!!!!!!!!!!!!!!!!!!#!!!!!!!!!!!!!!!!!!!!!!!!!
!
!
O;!)4!J.%%!?*1J*!;I';!)*!;I.!8)*'(0!,'4.5!)<.<!21(! & ! C# 5!;I.!)-./L1;.*;!&.*.(';1(!12!41/.!
,0,%),!,1-.! ( !12!%.*&;I! ' !)4!;I.!4$/!12!L1%0*1/)'%4! ? 6 G + H <!3$,I!'!L1%0*1/)'%! ? 6 G +H !
,1((.4L1*-4!;1!'!,0,%1;1/),!,14.;! ( 6 !/1-! ' 5!J);I!(.4L.,;!;1!C5!!)*!;I.!21%%1J)*&!J'0!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ? 6 G +H ! 7 + $ <!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GCYH!
$$( 6
O;!,'*!.'4)%0!8.!L(1:.*!G,2<!'%41!ZR5V[H!;I';!;I)4!L(1L.(;0!I1%-4!/1(.!&.*.('%%0!21(!,0,%),!
,1-.4!1:.(!'!2)*);.!2).%-! !) G&H 5!)2! ( 6 !4;'*-4!21(!'!,14.;!/1-! ' !J);I!(.4L.,;!;1! & 5!)<.<!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ( 6 ! P65 &65<<<<<<<5 & E6 &" Q <!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GC"H!
!
!
Theorem 19!
J:%, ( ,C:,;,?7?8$?,?@=:,@B,8:'9%3,',@F:-,%3:,B$:8=, !) G&H /,D$%3, G'5 &H ! " /,;'=,8:%, :G +H ,C:,
$%6!$=:E*@%:'%,9:':-;%@-1,L3:',@':,?;',D-$%:, :G +H ! 7 9 6 ?6 G +H /,D3:-:, 6 ,->'6,%3-@>93,,
6
%3:,6:%,@B,$'=$?:6,@B,?7?8@%@E$?,?@6:%6,E@=, ' /,;'=,D$%3, 9 6 $ !) G&H 1,!
!
Proof!
O2! :G +H )4!'*!)-./L1;.*;!&.*.(';1(!12! ( 5!;I.*!80!-.2)*);)1*! :G +H C ! :G +H 5!'*-!I.*,.!
:G +H 4 ! :G +H 21(!'%%!L14);):.!)*;.&.(4! 4 <!^.!J();.! :G +H ! 7 9$ +$ 5! 9$ $ !) G&H 5!'*-!J.!
$
;'?.! 4 ]! & 5!0).%-)*&! :G +H ! 7 9$ + ! 7 9$ + ! 7 9 M + <!O2! $ $ ( 6 !21(!41/.! 6 5!;I.*!
&
&
$
&$
&$
$
M
M
'%41! &$ $ ( 6 <!f$;;)*&! M ! &$ !)*!;I.!()&I;!I'*-!4)-.!12!;I.!'81:.!.N$';)1*5!&):.4! 9 &$ ! 9$ <!
6I)4!)/L%).4!;I.!4;';./.*;<!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#!
!
!
6I.!'81:.!;I.1(./!)4!,%.'(%0!)%%$4;(';.-!80!;I.!)-./L1;.*;!&.*.(';1(4!-.():.-!)*!;I.!
L(.:)1$4!.M'/L%.4<!!!!
!
!
!
!
!
!
"W
!
!
3. A matrix determining all primitive idempotent generators
!
^.!*1J!4I'%%!-.:.%1L!'!21$(;I!/.;I1-!;1!-.():.!;I.!)-./L1;.*;!&.*.(';1(!12!'!,0,%),!
,1-.!12!%.*&;I!'1!^.!,1*4)-.(!;I.!,1/L%.;.!2',;1()E';)1*!12!! + ' & " !)*;1!)((.-$,)8%.!
L1%0*1/)'%4!1:.(! !) G&H 5! G'5 &H ! " 5!JI),I!)4!J();;.*!'4!
!
8
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! + ' & " ! + 2$ G +H 5!!! 2Y G +H ! " <!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GCCH!!!
$ !Y
!
\.;! . 8.!'&')*!'!L()/);):.! '%3 !(11;!12!$*);0!)*!41/.!.M;.*4)1*!2).%-!12! !) G&H 5!-.2)*.-!'4!
'!E.(1!12!1*.!12!;I.!L1%0*1/)'%4!)*!;I.!(I4!12!GCCH!JI),I!J.!4I'%%!'44$/.!;1!8.! 2" G +H <!!
6I.*! . 5 . & 5<<<<<5 . &
E"&"
!'(.!;I.! E" !E.(14!12! 2" G +H 5!JI.(.! E" !)4!;I.!-.&(..!12! 2" G +H !O*!
E6 &"
&.*.('%5!;I.!)((.-$,)8%.!L1%0*1/)'%! 26 G+H !I'4!E.(14! . 6 5 . 6& 5<<<<<<<<5 . 6& 5!JI.(.! E6 !)4!;I.!
-.&(..!12! 26 G +H !315!;I.(.!)4!'!1*.c;1c1*.!,1((.4L1*-.*,.!8.;J..*!!;I.!,0,%1;1/),!,14.;4!!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (6 ! P65 6&5<<<<<<5 6& E6 &"Q !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GCRH!
!
'*-!;I.!L1%0*1/)'%4! 26 G +H !JI),I!J.!J();.!.ML%),);%0!'4!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 26 G +H ! + E6 5 *6 5" + E6 &" 5 <<<<<<<< 5 *6<E6 <!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GCSH!
!
^.!'-1L;!;I.!,1*:.*;)1*!;I';!;I.!)*-.M!6,4;'*-4!21(!;I.!%.'4;!)*;.&.(!12!;I.!(.%.:'*;!,14.;<!
61&.;I.(!;I.4.!)*-),.4!,1*4;);$;.!'*!)*-.M!4.;!O1!!
e*!;I.!1;I.(!I'*-5!;I.(.!)4!'%41!'!1*.c;1c1*.!,1((.4L1*-.*,.!8.;J..*!;I.!,0,%1;1/),!
,14.;4! (6 !'*-!;I.!L1%0*1/)'%4!G?7?8@'@E$;86H ?6 G +H $ "'& 5! 6 $ O 5!JI),I!'(.!-.2)*.-!'4!
!
E6 &"
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ?6 G +H ! + 6 5 + 6& 5 <<<<<<< 5 + 6& <!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GCVH!
!
`1(!'*0! $ $ P"5 C5<<<<<<<5 'Q !'*-!21(!'*0! 6 $ O 5!J.!-.2)*.!;I.!G/$%;)H4.;!!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (6G$ H ! P$65 $6&5<<<<<<<5 $6& E6 &"Q 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GCWH!
!
,1*;')*)*&!'%%!L(1-$,;4! $; 5!!J);I! ; $ (6 <!O;!J)%%!8.!,%.'(!;I';!'%%!.%./.*;4!)*! (6G$ H !8.%1*&!;1!
;I.!4'/.!,0,%1;1/),!,14.;5!8.!);!;I';!;I.4.!.%./.*;4!/'0!1,,$(!/1(.!;I'*!1*,.<!^.!4I'%%!
-.*1;.!;I)4!,0,%1;1/),!,14.;!80!;I.!)*-.M! $6 5!;I1$&I!;I)4!)4!*1;!*.,.44'()%0!;I.!%.'4;!
)*;.&.(!)*!;I.!,14.;<!O2!;I.!/$%;)4.;! (6G$ H !,1*;')*4!.',I!.%./.*;!12! ($6 !;,;)/.45!J.!4I'%%!
J();.! (6G $ H ! ;($6 <!=1(.!L(.,)4.%05!J.!I':.!;I.!21%%1J)*&!%.//'<!
!
!
"U
!
!
Lemma 20
)@-,;88, $ $ P"5 C5<<<<5 'Q ,;'=,;88, 6 $ O ,@':,3;6,
G)H!! (6G$ H ! '6$ ($6 ,B@-,6@E:,*@6$%$F:,$'%:9:-, '6$ ,D3$?3,=$F$=:6, E6 P,
G))H!! ?6 G. $ H ! & '6$ *$6 5" <!
!
Proof!
G)H!!K',I!.%./.*;!12! (6G$ H !,'*!8.!18;')*.-!2(1/! $6 !80!'!/$%;)L%),';)1*!J);I!41/.!L1J.(!12!
&5!'*-!.',I!.%./.*;!1,,$(4! '6$ ]! ! (6 !a! ($6 !_ E6 a E$6 !;)/.4<!
G))H!!#0!-.2)*);)1*! 2$6 G. $6 H ! Y 5!'*-!I.*,.! 2$6 G. $6& H ! <<<<<< ! 2$6 G. $6&
?6 G. $ H ! . $6 5 . $6& 5 <<<<< 5 .
$6& E6 &"
E$6 &"
H ! Y <!=1(.1:.(5!
! '6$ ?$6 G. H <!!315! ?6 G. $ H ! & '6$ *$6 5" <!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#!
!
!
O*!;I.!*.M;5!J.!,1*4)-.(! '6$ !'4!'*!.%./.*;!12! !) G&H 5!'*-!J.!)*;(1-$,.!'*!!O!: !O!c!
/';()M!I,!J);I!.%./.*;4!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ;$ 5 6 !]_! & '6$ *$6 5" 5!! $5 6 $ O <!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GCUH!
!
O*!L'(;),$%'(5!J.!I':.!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ;Y56 ! E6 !!'*-!! ;$ 5Y ! " <!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GCTH!
!
^.!'%41!)*;(1-$,.!;J1!/1(.!1L.(';)1*4!1*!;I.!2'/)%0!12!,0,%1;1/),!,14.;4!21(!,.(;')*!
2)M.-!:'%$.4!12!'!'*-,&1!O*!;I.!2)(4;!L%',.,!J.!-.2)*.!;I.!4$/!12!;J1!,14.;4!!!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ($ 5 ( M ! P; 5 C ! ; $ ($ 5 C $ ( M Q 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GCXH!!
!
'*-!4.,1*-%05!!;I.!L(1-$,;!!!
!
!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ($ ( M ! P;C ! ; $ ($ 5 C $ ( M Q 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GRYH!!!!!!!!!
!
JI.(.!;I.!4$/4!'*-!L(1-$,;4!12!;I.!)*;.&.(4!;,'*-!C,'(.!;'?.*!/1-!'1!
O;!J)%%!8.!18:)1$4!;I';!;I.!!/$%;)4.;!)*!;I.!(I4!12!GCXH!)4!'!$*)1*!12!,0,%1;1/),!,14.;45!
JI.(.!'!,14.;!/'0!1,,$(!/1(.!;I'*!1*,.<!d.*,.5!!1*.!,'*!J();.!)*!&.*.('%!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ($ 5 ( M ! ! ;8 (8 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GR"H!!
8
21(!,.(;')*!)*;.&.(4! ;8 !JI),I!-.L.*-!1*!$,'*-!M1,6I.!4'/.!,'*!8.!4')-!'81$;!;I.!L(1-$,;!
GRYH5!41!
!
!
"T
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ($ ( M ! ! C8 (8 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GRCH!
8
!
21(!,.(;')*!)*;.&.(4! C8 1!+%%!)*-),.4!)*!GR"H!'*-!GRCH!'(.!.%./.*;4!12!O<!
6I.!21%%1J)*&!4)/L%.!L(1L.(;).4!I1%-<!
!
Lemma 21
J:%,K,C:,%3:,B;E$87,@B,>'$@'6,@B,?7?8@%@E$?,?@6:%6,B@-,B$+:=,F;8>:6,@B,',;'=,&1,L3:',
G)H!,,K,$6,?8@6:=,>'=:-,%3:,@*:-;%$@'6,;6,=:B$':=,$',GR"H!!;'=,,GRCHg!
G))H!!,%3:,$'%:9:- ;Y ,$',GR"!H!,$6,:&>;8,%@, E$ ,$B, M ! &$ /,;'=,%@,,Y!!$B, M 4 &$ ,
G)))H!!%3:,6:%,@B,*@87'@E$;86, ? 6 G+H /, 6 $ O /,$6,?8@6:=,$' "'& ,>'=:-,E>8%$*8$?;%$@'g!
G):H!! ? 6 G + $ H ! ' 6$ ?$6 G +H g!
G:H!!! ?$ G +H? M G +H ! 7 ;8 ?8 G +H !D$%3,%3:,$'%:9:-6! ;8 !B-@E!GR"H5!;'=! ; Y ! E$ !$B, M ! &$ ;'=!
8
!!!!!!! ; Y ! Y !@%3:-D$6:g!
G:)H!!!$B, . $6,;,*-$E$%$F:,' %3 ,-@@%,@B,>'$%7/,%3:',, 7 ? 4 G. $ H ,$6,:&>;8,%@,Q,B@-, $ 4 Y /,;'=,%@,'!
4
!!!!!!!!!B@-,$GQ1,
!
!
^.!'(.!*1J!(.'-0!;1!21(/$%';.!'!;I.1(./!JI),I!L(1:)-.4!$4!J);I!'*1;I.(!/.;I1-!;1!
-.;.(/)*.!;I.!)-./L1;.*;!&.*.(';1(!12!;I.!,0,%),!,1-.!&.*.(';.-!80!'*!)((.-$,)8%.!
L1%0*1/)'%! 2$ G +H !2(1/!GCCH<!O*!;I.!21(/$%';)1*!12!;I)4!;I.1(./!J.!)*;(1-$,.!:.,;1(4!!
! < $ 5! $ $ O 5!1:.(! !) G&H !'*-!12!%.*&;I!!3!5!JI),I!I':.!'!1*.!1*!L14);)1*!$,'*-!E.(14!1*!
'%%!L14);)1*4!12!OlP$Q<,!
!
Theorem 22
J:%,%3:,$'=$?:6,@B,%3:,$--:=>?$C8:,*@87'@E$;86,$',GCCH!C:,6>?3,%3;%,,%3:,*-$E$%$F:, '%3 ,-@@%,
@B,>'$%7, . /,D3$?3,8$:6,$',6@E:,:+%:'6$@',B$:8=,@B, !) G&H /, G'5 &H ! " /,$6,=:B$':=,;6,;,<:-@,@B,
%3:,$--:=>?$C8:,*@87'@E$;8, 2" G +H 1,J:%,B>-%3:-E@-:, ( '>5& 5% ,C:,%3:,?7?8$?,?@=:,@B,8:'9%3,',
9:':-;%:=,C7,%3:,$--:=>?$C8:,*@87'@E$;8 2> G+H / > $ O /,;86@,B-@E,GCC!H/,;'=,
8:% :> G +H ! 7 9 6 ? 6 G +H /, 9 6 $ !) G&H /,C:,$%6,$=:E*@%:'%,9:':-;%@-1,L3:',%3:,?@:BB$?$:'%6, 9 6 ,
6$O
;-:,>'$&>:87,=:%:-E$':=,C7,%3:,6:%,@B,,8$':;-,:&>;%$@'6 7 ; $ 5 6 9 6 ! :> G. $ H /, $ $ O /,D$%3,
6$O
:> G. H ! Y ,B@-, $ ! > ,;'=, :> G. H ! " ,B@-, $ 4 > /,@-,:&>$F;8:'%87,$',E;%-$+,B@-E/,,C7,
I9 ! = > ,D$%3,?@8>E',F:?%@-6, 9 ,;'=, = > ! " & < > ,@B,8:'9%3,!O!1,
,
Proof
`(1/!6I.1(./!W!J.!?*1J!;I';!;I.!4.;!12!.N$';)1*4! :> G. $ H ! Y !21(! $ ! > !'*-! :> G. $ H ! " !
21( $ 4 > 5!I'4!'!41%$;)1*!'*-!4$,I!'!41%$;)1*!/$4;!8.!$*)N$.<!3$84;);$;)*&!;I.!(.%';)1*!12!
\.//'!CY!G))H!'*-!$4)*&!GCUH!&):.4!;I.!%)*.'(!4.;!12!.N$';)1*4!'4!4;';.-!)*!;I.!6I.1(./<!# !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
$
!
$
"X
!
!
Example 23!
6'?.! ' ! "V !'*-! & ! C !G,2<!'%41!KM'/L%.!"UH<!
6I.!,0,%1;1/),!,14.;4!'(.!!
!
(Y ! PYQ 5! (" ! P"5 C5 S5TQ 5! (R ! PR5 W5"C5XQ 5! (V ! PV5"YQ 5! (U ! PU5"S5"R5""Q <!
!
6I.!)*-.M!4.;! O ! PY5"5R5V5 UQ <!
6I.!1*%0! '6$ !I':)*&!'!:'%$.! 4 " !'(.! '"Y ! 'RY ! 'UY ! S !'*-! 'VY ! 'VR ! C <!
71((.4L1*-)*&!;1!;I.!'81:.!,0,%1;1/),!,14.;4!J.!I':.!;I.!L1%0*1/)'%4!!
!
?Y G +H ! " 5! ?" G +H ! + 5 + C 5 + S 5 +T 5! ?R G +H ! +R 5 + W 5 +"C 5 +X 5! ?V G +H ! + V 5 +"Y 5!
?U G +H ! + U 5 +"S 5 +"R 5 +"" <!
!
6I.!2',;1()E';)1*!)*!GCC!H!)4!&):.*!80! +"V & " ! + 26 G +H !J);I!
6$O
2Y G +H ! + 5 " 5 2" G +H ! + S 5 + 5 " 5 2R G +H ! + S 5 +R 5 " 5 2V G +H ! + C 5 + 5 " 5
2U G +H ! + S 5 +R 5 + C 5 + 5 " <!
!
^.!-.2)*.!'!L()/);):.!!"V %3 !(11;!12!$*);0! . !80!(.N$)()*&! 2" G. H ! Y <!
6I.!)-./L1;.*;!&.*.(';1(!12!;I.!,1-.! (" !,'*!8.!J();;.*!'4! :" G +H ! 7 9 6 ?6 G +H <!
6$O
+,,1(-)*&!;1!6I.1(./!CC5!J.!I':.!;I.!21%%1J)*&!4.;!12!.N$';)1*4!21(!;I.! 9 6 !
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! "9 Y 5 Y9" 5 Y9R 5 Y9V 5 Y9 U ! :" G. Y H ! " 5!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! "9 Y 5 Y9" 5 "9R 5 "9V 5 "9 U ! :" G. " H ! Y 5!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! "9 Y 5 "9" 5 "9R 5 Y9V 5 "9 U ! :" G. R H ! " 5!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! "9 Y 5 Y9" 5 Y9R 5 "9V 5 Y9 U ! :" G. V H ! " 5!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! "9 Y 5 "9" 5 "9R 5 "9V 5 Y9 U ! :" G. U H ! " <!
!
!
O*!/';()M!21(/!J.!J();.!;I)4!%)*.'(!4.;!12!.N$';)1*4!'4!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!I 9 = =" .
J);I!;I.!,1%$/*!:.,;1(4! 9 ! G9 Y 5 9" 5 9R 5 9V 5 9 U HL !'*-! =" ! G"5 Y5"5"5"HL !'*-!;I.!/';()M!
!
!
CY
>" Y Y Y Y ?
@
A
@" Y " " " A
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,I,G, @" " " Y " A <!
@
A
@" Y Y " Y A
@" " " " Y A
B
C
!
3)*,.!!;I.!)-./L1;.*;!&.*.(';1(!)4!$*)N$.5!;I)4!404;./!I'4!L(.,)4.%0!1*.!41%$;)1*!JI),I!
;$(*4!1$;!!;1!8.! 9 ! G"5"5"5 Y5 YH % <!!O2!J.!;'?.!21(!;I.!(I4!12!;I.!/';()M!.N$';)1*!;I.!:.,;1(!
= R ! G"5"5 Y5"5"H% 5!JI),I!,1((.4L1*-4!;1!;I.!L1%0*1/)'%! 2R G +H 5!J.!2)*-! 9 ! G"5 Y5"5 Y5"HL !'4!
41%$;)1*<!d.*,.5!21(!;I.!)-./L1;.*;!&.*.(';1(4!12!;I.!;J1!!"#,1-.45!J.!2)*-!!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! :" G +H ! " 5 + 5 + C 5 + R 5 + S 5 + W 5 +T 5 + X 5 +"C 5!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! :R G +H ! " 5 +R 5 + W 5 + U 5 +X 5 +"" 5 +"C 5 +"R 5 +"S <!
!
^.!:.()2).-!;I';!;I.!4'/.!.ML(.44)1*4!'(.!-.%):.(.-!80!.N<!G"WH5!$4)*&!;I.!,I.,?!
L1%0*1/)'%4!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 3" G +H ! +"" 5 +T 5 + U 5 +V 5 +R 5 + C 5 + 5 " 5!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 3R G +H ! +"" 5 +"Y 5 +X 5 +T 5 + W 5 + S 5 +R 5 " <!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!# !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! !
!
!
Example 24!
+&')*!J.!4;$-0!;I.!,'4.! ' ! "Y !'*-! & ! R !G,2<!KM'/L%.!"TH<!6I.!,0,%1;1/),!,14.;4!'(.!
(Y ! PYQ 5! (" ! P"5R5X5 UQ 5! (C ! PC5 W5T5 SQ !5! (V ! PVQ 5!'*-!;I.!,1((.4L1*-)*&!)((.-$,)8%.!
L1%0*1/)'%4! 2Y G +H ! + & " 5! 2" G +H ! + S & +R 5 + C & + 5 " 5! 2C G +H ! + S 5 +R 5 + C 5 + 5 " 5!
"
<!
2V G +H ! + 5 " <!^.!?*1J!;I';! '"Y G +H ! 2" G +H !)4!)((.-$,)8%.!'*-!;I';! 2" G +H !&.*.(';.4! ("Y5R5"
6I.!,1.22),).*;4! 9$ 5!! $ $ O ! PY5"5 C5VQ 5!,'*!8.!-.;.(/)*.-!80!;I.!/';()M!.N$';)1*!
!
>" " " "? > 9 Y ? > " ?
@
A@ A @ A
" " &" &"A @ 9" A @ Y A
<!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! I 9 ! !! @
!
@" &" &" "A @ 9 C A @ " A
@
A@ A @ A
B" &" " &"C B 9V C B " C
!
^.!2)*-! 9 ! GY5 &"5"5"HL 5!'*-!41!;I.!)-./L1;.*;!&.*.(';1(!12!;I.!!"#,1-.! (" !)4!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!! :" G +H ! & + 5 + C & +R 5 + S 5 +V 5 + W & + U 5 +T & +X <!
!
C"
!
^.!18;')*.-!;I.!4'/.!.ML(.44)1*!80!'LL%0)*&!G"WH!J);I! 3" G +H ! + W 5 + V & + & " <!
!
b1J5!J.!;'?.! & ! X <!6I.!,0,%1;1/),!,14.;4! (" !'*-! (R !4L%);!'*-!J.!I':.!*.J!,14.;4!
(Y ! PYQ 5! (" ! P"5XQ 5! (C ! PC5TQ 5! (R ! PR5 UQ 5! (S ! PS5 WQ 5! (V ! PVQ !'*-! O ! PY5"5 C5R5 S5VQ <!
94)*&!;I.!4'/.!)*-),.45!J.!I':.!;I.!21%%1J)*&!)((.-$,)8%.!L1%0*1/)'%4! 2Y G +H ! + & " !J);I!
E.(1 . Y 5 2" G +H ! + C 5 6 + 5 " J);I!E.(14! . 5 . X 5 2C G +H ! + C 5 G6 5 "H + 5 " !J);I!E.(14! . C 5 . T !5!
2R G +H ! + C & G6 5 "H + 5 " !J);I!E.(14! . R 5 . U 5! 2S G +H ! + C & 6 + 5 " !J);I!E.(14! . S 5 . W !'*-!
2V G +H ! + 5 " !J);I!E.(1! . V <!!d.(.5!;I.!.%./.*;! 6 $ !) GXH !4';)42).4! 6 C 5 6 & " !_!Y<!
6I.!/';()M!I,!)*!;I)4!,'4.!,'*!(.'-)%0!8.!-.;.(/)*.-!'*-!)4!.N$'%!;1!!
!
&"
&"
&"
&" "?
>"
@
6 &"AA
&6 &6 & " 6 5 "
@"
@" &6 & "
6
6 &6 & " "A
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! I ! @
A<!
&6 &6 & " &"A
6
@" 6 5 "
@"
"A
6 &6 & " &6 & "
6
@@
A
"
"
&"
&" &"CA
B"
!
6I.!.N$';)1*! I 9 ! =" !J);I! 9 ! G9 Y 5 9" 5 9 C 5 9R 5 9 S 5 9V HL !'*-! =" ! G"5 Y5"5"5"5"HL !0).%-4!;I.!
41%$;)1*! 9 ! G&"5 6 5 6 5 "5 &6 & "5 &6 5 &"HL 5!JI),I!)*-..-!,1((.4L1*-4!;1!;I.!)-./L1;.*;!
&.*.(';1(! :G"H G +H !'4!,1/L$;.-!)*!KM'/L%.!"T<!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!
!
!
>$.!;1!;I.!L(.:)1$4!.M'/L%.4!;I.!21%%1J)*&!;I.1(./!L(.4.*;4!);4.%2<!
!
Theorem 25!
J:%, 2$" G +H5 2$C G +H5<<<<<<5 2$8 G +H /,D$%3, $" 5 $C 5<<<<<5 $8 $ O /,C:,8,$--:=>?$C8:,*@87'@E$;86,$'%-@=>?:=,
$',,GCH/,;'=,8:%,(,C:,%3:,?7?8$?,?@=:,9:':-;%:=,C7,%3:,*-@=>?%, 2G +H ,@B,%3:6:,*@87'@E$;86<!
L3:',%3:,$=:E*@%:'%,9:':-;%@-, :G +H ! 7 9 6 ?6 G +H @B,(,$6,=:%:-E$':=,C7,%3:,6:%,@B,8$':;-,
6$O
:&>;%$@'6, I 9 ! = /,D3:-:, = ,$6,%3:,?@8>E',F:?%@-,@B,8:'9%3,!O!,D$%3,<:-@6,@',%3:,
*@6$%$@'6, $" 5 $C 5<<<<<5 $8 /!;'=,@':6,:86:D3:-:<!
!
!
O*!L'(;),$%'(5!1*.!18;')*4!;I.!*-$E$%$F:,$=:E*@%:'%,9:':-;%@-65!!)<.<!;I.!)-./L1;.*;!
&.*.(';1(4!12!;I.!41c,'%%.-!E$'$E;8,1(!$--:=>?$C8:,?@=:6!G,2<!.<&<!ZV[H5!!80!;'?)*&!!21(! = !'!
:.,;1(!J);I!@$4;!1*.!1*.!'*-!E.(14!1*!'%%!1;I.(!L14);)1*45!JI.(.'4!;I.!)-./L1;.*;!
&.*.(';1(4!21(!;I.!E;+$E;8!?@=:6!'(.!18;')*.-!80!;'?)*&!21(! = !'!:.,;1(!J);I!@$4;!1*.!E.(1!
'*-!1*.4!1*!'%%!1;I.(!L14);)1*4<!
!
!
CC
O;!;$(*4!1$;!;I';!;I.!)*:.(4.!12!;I.!/';()M! I ,'*!.'4)%0!8.!-.;.(/)*.-<!
!
Theorem 26!
J:%,I,C:,%3:,E;%-$+,%3:,,:8:E:'%6,@B,D3$?3,;-:,=:B$':=,,$',GCUH!!;'=,8:%, I 2 C:,%3:,E;%-$+,,,
@C%;$':=,B-@E!I,!C7,$'%:-?3;'9$'9,%3:,?@8>E'6,$'=:+:=,C7,6,;'=, & 6 /,B@-,;88, 6 $ O 1,,L3:',
I &" ! ' &"I 2 /,D3:-:,,%3:,$'F:-6:,@B,',$6,%@,C:,%;4:',$', !) G&H 1!
!
Proof
^.!,1*4)-.(!;I.!)**.(!L(1-$,;!12!(1J!$,!'*-!,1%$/*!6,12!;I.!/';()M!I1!94)*&!GCUH!'*-!
\.//'!CY!G))H5!J.!&.;!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 7 ;$ 5 4 ;4 5 6 ! 7 ?4 G. $ H?6 G. 4 H _!
4$O
4$O
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 7 G. 5 .
$4
$4&
5 <<<< 5 . $4&
E4 &"
HG. 64 5 . 64& 5 <<<< 5 . 64&
E6 &"
H _!
4$O
E6 &" E4 &"
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 7 7 G 7 . G $&
4$O ; ! Y
C
5 6& ; H 4
C !Y
E6 &" E4 &"
H _ 7 7 G 7 . G $ 5 6&
4$O ; ! Y
; &C
H 4& C
H <!
C !Y
!
O*!1(-.(!;1!.:'%$';.!;I.!(I4!12!;I.!%'4;!.N$'%);05!J.!4I'%%!/'?.!$4.!12!;I.!2',;!;I';!21(!
.:.(0!2)M.-!:'%$.!12!C5!!;I.!4.;!P6& ; &C ! Y * ; * E6 & "Q !)4!.N$'%!;1!;I.!,0,%1;1/),!,14.;!
(6 ! P6& ? ! Y * ? * E6 & "Q <!d.*,.5!J.!18;')*!
E6 &" E4 &"
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 7 7 G 7 . G $ 5 6&
4$O ? ! Y
?
H 4& C
H _!
C !Y
E6 &"
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 7 Z7 G. i 4 5 . i 4& 5<<<<<<< 5 . i 4&
? ! Y 4$O
E 4 &"
E6 &" ' &"
H[ _! 7 Z7 . i8 [5!
? !Y 8 !Y
!
?
JI.(.! . i ! . $ 5 6& !)4!41/.!G*1;!*.,.44'()%0!L()/);):.H! ' %3 !(11;!12!$*);0<!`1(! 6 4 &$ !1*.!I'4!
;I';! . i 4 " 5!'*-!I.*,.!;I.!.ML(.44)1*!8.;J..*!8(',?.;4!)4!.N$'%!;1!Y!21(!.',I!:'%$.!12! ? <!
`1(! 6 ! &$ ;I)4!.ML(.44)1*!)4!'%41!Y5!.M,.L;!21(! ? ! Y 5!&):)*&! . i Y 5. i Y 5<<<< 5 . i Y ! ' <!!!!!!!!!
d.*,.5!J.!/'0!,1*,%$-.!;I';! 7 ; $ 54 ; 4 5 6 ! Y 5!21(! 6 4 &$ !'*-! 7 ;$ 5 4 ;4 5 &$ ! ' !21( 6 ! &$ <#!!!!!!!!
4$O
4$O
!
Corollary 27
L3:,*-$E$%$F:,$=:E*@%:'%,9:':-;%@-6, D> G +H ! 7 9> 5 6 ?6 G +H /, > $ O /,@B,%3:,E$'$E;8,?7?8$?,
6$O
?@=:6,B@-,?:-%;$',F;8>:6,@B, ' ;'=, & /,;-:,=:%:-E$':=,C7,,%3:,F:?%@-, 9> G, ' &" I 2 < > /,D3:-:,
< > ,$6,%3:,F:?%@-,@F:-, !) G&H ,@B,8:'9%3,!O!,D$%3,;,@':,@',*@6$%$@',>,;'=,<:-@6!:86:D3:-:1,
L3:,$=:E*@%:'%,9:':-;%@-6, E> G +H ! 7 9 i> 5 6 ?6 G +H D3$?3,9:':-;%:,%3:,,E;+$E;8!!?7?8$?!
&"
6$O
2
?@=:6/,;-:,=:%:-E$':=,C7, 9 i> ! ' I = > /,D$%3, = > G, " & < > 5,,;'=,D3:-:,"!,6%;'=6,B@-,%3:,
;88#@':,F:?%@-,@B,8:'9%3,!O!1!
!
!
CR
6I)4!,1(1%%'(0!)/L%).4!;I';!;I.!L()/);):.!)-./L1;.*;!&.*.(';1(!!12!;I.!,0,%),!,1-.!
&.*.(';.-!80! + 2$ G +H 5! > $ O 5!,1((.4L1*-4!;1!;I.!,1%$/*!12!I!J);I!)*-.M!! & > 5!
$ 4>
1,,'4)1*'%%0!,'%%.-! ;& > <!`1(!L(',;),'%!(.'41*45!J.!-.*1;.!;I)4!)-./L1;.*;!&.*.(';1(!80!
D>j G +H 5!'*-!41!J.!I':.!21(!'%%! > $ O !
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! D>j G +H ! D & > G +H ! ' &" 7 ; 6 5 &> ?6 G +H ! ' &" 7 G ;& > H 6 ?6 G +H !!!!!!!!!!!!!!!!!!!!!!!GRRH!!!!!!!!!
6$O
6$O
'*-!4)/)%'(%05!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! E>j G +H ! " & D>j G +H ! " & D & > G +H <!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!GRSH !!!!!!!!!!!!!!!!!!!!!!!!!!!!
!
!
O*!;I.!*.M;!;I.1(./!J.!4;';.!!41/.!4)/L%.!40//.;(0!L(1L.(;).4!12!;I.!/';()M!I1!61!-1!
;I)4!)*!'!,1*,)4.!J'05!J.!4I'%%!/'?.!$4.!12!;I.!m(1*.,?.(!40/81%! = ; 5C !JI),I!4;'*-4!21(!"!
)2! ; ! C 5!'*-!21(!Y!)2! ; 4 C <!
!
Theorem 28
AB,D:,=:'@%:,%3:,:8:E:'%6,@B,%3:,E;%-$+,I,C7 ;$ 5 M /, $5 M $ O /,%3:',%3:,B@88@D$'9,-:8;%$@'6,
3@8=,$', !) G&H ],
G)H!!! E$ ;$ 5 M ! E M ; M 5$ g!
G))H!! 7 ;$ 54 ;4 < M ! '= $ 5 & M g!
4
G)))H!!! 7 E4 ; 4 5$ ; 4 5 M ! 'E$ = $ 5 & M g !
4
G):H!! 7 E4 &";$ 5 4 ; M 54 ! 'E$ &"= $ 5 & M <!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
4
,
Proof!
KN$'%);0!G)H!21%%1J4!)//.-)';.%0!2(1/!;I.!-.2)*);)1*!GRUH!12!;I.!.%./.*;4!12!I1!
6I.!1;I.(!(.%';)1*4!'(.!)//.-)';.!,1*4.N$.*,.4!12!G)H!'*-!6I.1(./!CW!!!<!!!!!!!!!!!!!!!!!!!!!!!!#!
!
Remark 29!
6I.!)*:.(4.4!)*!6I.1(./!CT!G):H5!'(.!1*%0!-.2)*.-!'4!%1*&!'4! E4 !'*-! E$ !'(.!$*.N$'%!;1!
E.(1!)* !) G&H <!=$%;)L%0)*&!81;I!4)-.4!12!(.%';)1*!G):H!80! E$ G! E M H &):.4!;I.!2',;1(!
E M ; M 54 a E4 )*!;I.!%I4!JI),I!'%J'04!.M)4;4<!
b1;),.!;I';!)*!;I.!'81:.!;I.1(./5!'%%!4$//';)1*4!'(.!1:.(!O5!'*-!;I';!'%%!)*;.&.(4! E8 !'*-!
'!'(.!;1!8.!;'?.*!)*! !) G&H <!!
!
!
Lemma 30
J:%, G'5 &H ! " ,;'=,8:%,-,C:,%3:,@-=:-,@B,&,E@=>8@,'1,
!
CS
G)H!!L3:,6$<:, E$ ,@B,%3:,?7?8@%@E$?,?@6:%,( $ ,$6,;,=$F$6@-,@B,-/,B@-,;88, $ $ O 1,
G))H!!AB, &" ,$6,;,&#*@D:-,E@=>8@,'5!%3:', (6 ! (& 6 ,B@-,;88, 6 $ O 1,
Proof
G)H!!6I.!.%./.*;4!12! (" !'(.!L(.,)4.%0!;I.!.%./.*;4!12!;I.!&(1$L! . ]!( & % !12!1(-.(!-5!'*-!
41! E" ! - <!6I.!,0,%1;1/),!,14.;! (6 !,1*4)4;4!12!;I.!.%./.*;4! P65 6&5<<<<<5 6& E6 &"Q 5!'*-!I.*,.!!
E6 !/$4;!8.!'!-):)41(!12! E" ! - <!!
G))H!!O2! &" ! & 8 5!;I.*! & 6 ! & 8 6 5!41!'%%!.%./.*;4!12! (& 6 !,'*!8.!J();;.*!'4!'!&cL1J.(<!3)*,.!
! (6 !_! (& 6 !5!;I.!(.%';)1*!)*!G))H!21%%1J4!)//.-)';.%0<!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!# !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
6I.!21%%1J)*&! & ';!%.'4;!)*!;I.!8)*'(0!,'4. & !J.%%c?*1J*!L(1L.(;).4!,'*!.'4)%0!8.!-.():.-!
)*!;I.!$4$'%!J'0!G,2<!ZR5V[H<!
!
!
Theorem 31
)@-,;'7,',;'=,*-$E:,*@D:-,&,D$%3, G'5 &H ! " /,@':,3;6,%3;%,%3:,*-$E$%$F:,$=:E*@%:'%,
9:':-;%@-6, D 6 G+H /, 6 $ O /,@B,?7?8$?,?@=:6,@B,8:'9%3,',@F:-,&,6;%$6B7]!
!
G)H!!! 7 D 6 G +H ! " g!
6$O
G))H!! D 6 G +HD > G +H ! Y !B@-,;88, 65 > $ O D$%3, 6 4 > g!
G)))H!! E> G +H ! " & D > G +H g!!!
G):H!!!$B, :G +H !$6,%3:,$=:E*@%:'%,9:':-;%@-,@B,6@E:,?7?8$?,?@=:/,%3:', :G +H ! 7 D 6 G +H ,B@-!!!
6$L
!!!!!!!!6@E:,6>C6:%,L , O g!
G:H!!!%3:,?@=:,D$%3,9:':-;%@-,*@87'@E$;8, 2$" G +H 2$C G +H<<<<<<<2$8 G +H ,3;6,$=:E*@%:'%,9:':-;%@-,,
,,,,,,,, " & D$j" G +H & D$jC G +H<<<<<<< & D$j8 G +H 1,
!
!
!
CV
!"#$%&'()*&+*#,&+&-.*)-/#)0#GR-1)%&/#
#
!"#$%&%'($n$)"$*+,"$-+*'#'."$'(#"&"/$%(0$q$*+,"$-/',"$-+1"/$*%#'*23'(&$ 4 n5 q 7 ! 8 6$
9"$:+(*'0"/$#;"$GR-:+0"*$ Cni 5q 5t 5$$ 8 " i " t 5$%*$0"2'("0$'($<":#'+($86$=#$1'>>$)"$:>"%/$#;%#$
#;"$">","(#*$+2$#;"$&/+?-$H:=<q>$4:26$<":#'+($87$%/"$#;"$*%,"$%*$#;"$">","(#*$+2$#;"$
:3:>+#+,':$:+*"#$ C8 ! A85 q566665 q r #8B 6$<','>%/>35$#;"$">","(#*$+2$#;"$:+*"#$ H i $@! xi H 4:26$
4C775$%/"$#;"$*%,"$%*$#;"$">","(#*$+2$#;"$:3:>+#+,':$:+*"#$ Ci $'2$ xi $ Ci 5$2+/$ 8 " i " % 6$
D;"/"2+/"5$1"$:%($'0"(#'23$ xi $1'#;$$#;"$>"%*#$">","(#$i$+2$$ Ci 6$E*$%$:+(*"F?"(:"5$#;"$
'(0"G$ i +2$ H i $(+1$/?(*$#;/+?&;$#;"$*"#$ S & U n 5$'6"6$#;"$*"#$+2$'(0':"*$+2$#;"$:3:>+#+,':$
:+*"#*$,+0$n$$1;':;$%/"$:+-/',"$1'#;$n5$%(0$1"$$1/'#"$2/+,$(+1$+($'(*#"%0$+2$4H7$
$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ' n 4 x7 ! ( Ps 4 x7 5$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$4IC7$$
s$S &U n
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
1;"/"$#;"$'//"0?:')>"$-+>3(+,'%>$ Ps 4 x 7 5$ s $ S 5$$'*$0"2'("0$)3$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ Ps 4 x7 ! ( 4 x # ) j 7 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$4IJ7$
j$Cs
$
=2$1"$#%K"$ K @! H $1;"($:+(*#/?:#'(&$#;"$:+0"$ Cni 5q 5t 5$1"$;%."$ % ! t $%(0$*+$ * 4 n7 ! rt 6$$
9'#;$#;'*$:;+':"$2+/$K5$#;"$GR-:+0"*$%/"$L,%G',%>M$:+0"*5$0"2'("0$)3$#;"$&"("/%#+/*$
g 4 i 7 4 x7 @! Pi 4 x 7 5$ i $ S & U n 6$D;"$'0",-+#"(#$&"("/%#+/$+2$ C ni 5q 5t $5$%*$2+>>+1*$2/+,$
N+/+>>%/3$HO5$:%($)"$1/'##"($%*$ e 4 i 7 4 x 7 ! , +i 5 j c j 4 x7 5$1;"/"$ + i 5 j $'*$#;"$ j th $:+,-+("(#$+2$
j$S
#;"$:+>?,($.":#+/$ +i ! - P # .# i $%(0$ .# i $'*$#;"$:+>?,($1'#;$'(0"G$ # i $+2$#;"$,%#/'G$ M 5$
%(0$;"(:"5$ +i 5 j ! / P5 j # . j 5 # i 6$
Q+1$1"$K(+1$#;%#$2+/$%(3$ i $ S & U n 5$#;"$.%>?"$+2 mi $'*$"F?%>$#+$ r @! +/0 n 4q 7 5$%(0$
;"(:"5$1"$;%."$2+/$*?:;$,%#/'G$">","(#*$ . j 5i $
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ . j 5i ! #
r
pij 58 5$$$$ i $ S & U n 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
mij
4IO7$
$
9"$/",%/K5$#;%#$ mij $'*$#;"$0"&/""$+2$#;"$-+>3(+,'%>$ Pij 4x7 5$1;'>"$ pij 58 $'*$#;"$:+"22':'"(#$
m #8
+2$ x ij $'($#;%#$*%,"$-+>3(+,'%>6$=($#;"$-/"*"(#$:%*"$+2$GR-:+0"*$1'#;$ Pi 4 x7 5$ i $ S & U n 5$
%*$&"("/%#+/5$1"$%>*+$K(+1$#;%#$$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$! S & U n !R$ * 4 n7 S r ! t 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$4IT7$
$
$
HJ
=($-%/#':?>%/5$'#$2+>>+1*$2/+,$4IO7$#;%#$#;"$:+>?,($.":#+/$ .8 $;%*$%*$ j th $:+,-+("(#$
$
r
p j 58 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$4IU7$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ . j 58 ! #
mj
V/+,$N+/+>>%/3$HO$$%(0$2/+,$#;"$2%:#$#;%#$ m# i ! mi $2+/$%>>$ i $ S 5$1"$,%3$:+(:>?0"$#;"$
2+>>+1'(&$/"*?>#6$
$
23&)-&'#45#
The idempotent generator of the GR-code Cni 5q 5t 5 i $ S & U n , is given by 0iW 4 x7 ! 8 # 1iW 4 x7 ,
with 1iW 4 x7 ! n #8 , #
j$S
r
p#ij 58c j 4 x7 6$$
mij
$
=($+/0"/$#+$0"#"/,'("$%>>$'0",-+#"(#$&"("/%#+/*$+2$#;"$ t $"F?'.%>"(#$GR-:+0"*$ C ni 5q 5t 5$
i $ S & U n 5$'#$'*$*?22':'"(#$#+$:+,-?#"$ 08 4 x7 5$+/$"F?'.%>"(#>3$#;"$:+>?,($.":#+/ .8 6$D;"$
:+,-+("(#*$+2$%(3$+#;"/$:+>?,($.":#+/$ .i $+2$M5$$1'#;$ i $ S & U n 5$:+(*#'#?#"$%$
-"/,?#%#'+($+2$#;"$:+,-+("(#*$+2$ .8 6$
$
$
#
23&)-&'#44##
Let .i 5 j , i5 j $ S , be the elements of the matrix M.
4'7$$If $j and k are elements of S & U n 5$and if$ l $U n $is such that$ lk ! j $mod$n, then$
.l 5k ! . j 58 X$
4''7$$The components of the column vector .k , k $ S & U n , form a permutation of the
components of$ .8 X$
4'''7$$The group U n S H acts transitively on the set of column vectors A .i8 , .iH , ….., .it B,
where i8 , iH , ……, it are the t elements of S & U n .
6-))0$
4'7$$V/+,$4HO7$1"$'(2"/$#;%#$ .l 5k ! #
r
r
plk 58 ! #
p j 58 ! . j 58 6$
mlk
mj
4''7$$=2$j$%(0$k$%/"$:+(*'0"/"0$%*$">","(#*$+2$#;"$&/+?-$ U n 5$#;"/"$'*$-/":'*">3$+("$">","(#$
l $ U n $1'#;$ lk ! j $,+0$ n 5$2+/$%(3$2'G"0$-%'/$j$%(0$k6$=2$ j $%(0$ k $%/"$">","(#*$+2$ S & U n $
:+(*'0"/"0$%*$&/+?-5$#;'*$?('F?"$">","(#$ l $'*$%>*+$'($ S & U n 6$<+5$2+/$%$2'G"0$'(0"G$
k $ S & U n 5$#;"$:+,-+("(#*$+2$ . k $1;':;$:+//"*-+(0$#+$">","(#*$+2$ S & U n 5$2+/,$%$
-"/,?#%#'+($+2$#;"$*%,"$K'(0$+2$:+,-+("(#*$+2$ .8 6$Q"G#5$#%K"$#;"$*?)*"#$ S a 2 S $1'#;$
#;"$-/+-"/#3$#;%#$ s $ S a $'2$%(0$+(>3$'2$ 4 n5 s 7 ! a 6$Y'.'0"$%>>$">","(#*$+2$ S a $)3$ a $%(0$
:+(*'0"/$#;"$F?+#'"(#*$%*$">","(#*$+2$ U n S a 6$Z3$*','>%/$%/&?,"(#*$%*$)"2+/"5$1"$:%($-/+."$
$
HO
(+1$#;%#$#;"$:+,-+("(#*$+2$ . k $1;':;$:+//"*-+(0$#+$">","(#*$+2$ S a 5$%/"$%$-"/,?#%#'+($
+2$#;"$*%,"$K'(0$+2$:+,-+("(#*$+2$ .8 6$N+(#'(?"$#;'*$-/+:"**$?(#'>$#;"/"$%/"$(+$
:+,-+("(#*$+2$ . k $>"2#6$Q+#':"$#;%#$ S8 ! S & U n $'($#;'*$:+(#"G#6$
4'''7$$D;'*$2+>>+1*$',,"0'%#">3$2/+,$4''75$1;"($%-->3'(&$#;"$2%:#$#;%#$ S & U n $:+(*'0"/"0$
%*$%$&/+?-5$'*$'*+,+/-;':$#+$#;"$F?+#'"(#$&/+?-$ U n S H 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$"$$$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
7&'.-8#4!$
=($+/0"/$#+$:+,-?#"$#;"$'0",-+#"(#$&"("/%#+/$ 08 4 x7 5$1"$,?*#$K(+1$#;"$:+"22':'"(#*$ p j 58 $
+2$%>>$'//"0?:')>"$-+>3(+,'%>*$ Pj 4 x 7 6$=($-/%:#':"5$#;"$*":+(0$,"#;+0$2+/$:+,-?#'(&$
'0",-+#"(#$&"("/%#+/*5$0'*:?**"0$'($#;"$-/".'+?*$*":#'+(5$*"",*$#+$)"$>"**$0",%(0'(&5$
*'(:"$+("$+(>3$;%*$#+$:+,-?#"$ h 487 4 x7 ! x n # 8S g 487 4 x7 $2+/$%$&'."($ g 487 4 x 7 @! P8 4 x7 6$
#
=($#;"$("G#$*":#'+(*$1"$*;%>>$0"/'."$:+(*#/?:#'+($/?>"*$2+/$#;"$'0",-+#"(#$&"("/%#+/*$+2$
&"("/%>'["0$/"*'0?"$:+0"*$'($#;"$:%*"*$n=p 5$ n ! H p 5$ n ! p m $%(0$ n ! H p m 6$9"$/",'(0$
#;"$/"%0"/$#;%#$#;"*"$:+0"*$%>>$)">+(&$#+$#;"$*?)2%,'>3$+2$t-/"*'0?"$:+0"*$4:26$\8]$%(0$
<":#'+($8$+2$#;'*$^"-+/#76$$$
#
9"##23&#case n ! p $$$$$$$$$
=($#;"$*-":'%>$:%*"$ n ! p $1;"/"$p '*$%($+00$-/',"5$+("$;%*$$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ x p # 8 ! 4 x # 87' p 4 x7 5$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
4_P7$
$
1'#;$0"&$ ' p 4 x7 $R$ p # 8 6$V?/#;"/,+/"5$ q $'*$*+,"$-/',"$-+1"/$1'#;$ 4 p5 q 7 ! 8 6$E>>$$
:3:>+#+,':$:+*"#*$ Ci $,+0 p 5$1'#;$/"*-":#$#+$ q 5$ i 3 P 5$;%."$#;"$*%,"$*'["$ r @! $+/0 p 4q 7 6$
`"(:"5$%>>$'//"0?:')>"$-+>3(+,'%>*$ Pi 4 x7 5$ i 3 P 5$%/"$+2$#;"$*%,"$0"&/""$r $%(0$
% @! * 4 n7 S r ! p # 8S r 6$N+(*"F?"(#>35$$%>>$,%#/'G$">","(#*$ . P 5i 5$ i 3 P5 $%/"$"F?%>$#+$r.$
a+/"+."/5$2+/$#;"$+#;"/$">","(#*$+2$M$+("$;%*$$
$
m
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ . j 5i ! # i pij 58 5$$ i5 j $ S SAPB 5$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$4_87$
mij
1;'>"$ . j 5 P ! 8 $2+/$%>>$j5 %*$%>1%3*6$
$
:;.'(<&#49$
D%K"$ n ! 8I $%(0$ q ! I 6$=#$2+>>+1*$#;%#$ r @! +/0 8I 4I7 ! I 5$%(0$#;%#$$#;"$:3:>+#+,':$:+*"#*$
,+0$8I$%/"$$
$$
$$$$$$$$$$$$$$$$$$$ CP ! APB 5$ C8 ! A85I5UB 5$ CH ! AH5 J5CB 5$ C_ ! A_58P58HB 5$ CO ! AO5T588B 6$
$
$
HT
V?/#;"/,+/"5$1"$1/'#" x8I # 8 ! PP 4 x7 P8 4 x7 PH 4 x7 P_ 4 x7 PO 4 x7 $1'#;$#;"$'//"0?:')>"$
-+>3(+,'%>*$
$
$$$$$$$$$$$ PP 4 x7 ! x # 8 5$ P8 4 x 7 ! x I # x # 8 5$ PH 4 x7 ! x I 4 x H 4 x # 8 5$ P_ 4 x7 ! x I 4 x H # 8 5$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ PO 4 x 7 ! x I # x H # x # 8 6$
$
Z3$%-->3'(&$"F6$4IO7$$1"$2'(0$#;"$2+>>+1'(&$,%#/'G$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$P$$$$$8$$$$$$H$$$$$$_$$$$$$O$
58 P P P P 6 P
7
8
78 P #8 #8 88 8
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ M ! 78 #8 #8 8 P 8 H 6$
7
8
78 #8 8 P #88 _
78 8 P #8 #88 O
9
:
$
<+5$1;"($?*'(&$#;%#$ n #8 ! 8 $,+0$I5$#;"$-/','#'."$'0",-+#"(#$&"("/%#+/$1;':;$:+//"*-+(0*$
#+$ C8 $'*$"F?%>$#+$
$
$$$$$$$$$$$$$$$$$$$$$$$ 18W 4 x7 ! , . s 5#8cs 4 x7 ! , . s 5_ cs 4 x7 $R #c8 4 x7 4 cH 4 x7 # cO 4 x7 $
s$S
s$S
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$R$ # x # x I # x U 4 x H 4 x J 4 x C # x O # xT # x88 6$
$
V'(%>>35$1"$+)#%'($
$
$$$$$$$$$$$$$$$$$$$$$$$ 08W 4 x 7 ! 8 # 18W 4 x 7 ! x88 4 x U 4 xT 4 x O # x J # x C 4 x I # x H 4 x 4 8 5$
$
1;':;$'*$'(0""0$#;"$*%,"$-+>3(+,'%>$%*$1"$2+?(0$'($bG%,->"$8J6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$"$$
$
$
:;.'(<&#4=$
9"$#%K"$ n ! I8 $%(0$ q ! H 6$
<'(:"$I8$'*$%$-/',"5$1"$;%."$#;%#$ U I8 $1'#;$">","(#*$ @8?H?""""""""" ?IP> $'*$%$:3:>':$&/+?-6$$
9"$2'(0$#;%#$ r A! +/0 I8 4H7$R$C5$%(0$;"(:"$ % ! * DI8C B r ! J 6$D;"$&/+?-$ H &"("/%#"0$)3$H$
:+(*'*#*$+2$#;"$">","(#*$ @8?H?_?T?8J> 6$=($#;'*$:%*"$H$'*$%$Ic$/"*'0?"5$*'(:"$ _ I ! H $,+0$I86$
=#$2+>>+1*$#;%#$%>>$">","(#*$+2$ H %/"$Ic/"*'0?"*6$V'/*#5$1"$"(?,"/%#"$%>>$*'G$:+*"#*$+2 H @$
$$
$$$$$$$$$$$$$ H 8 ! @8?H?_?T?8J> 5$$$$$$$$ H H ! @I?J?8H?8O?H_> 5$$$$$$ H I ! @C?U?8P?8T?HP> 5$$
$$$$$$$$$$$$$ H _ ! @O?8_?8U?HC?HT> 5$$ H C ! @88?8I?H8?HH?HJ> 5$$ H J ! @8C?HI?HO?HU?IP> 6$
$
$
HU
9"$/",%/K$#;%#$'2$1"$0"(+#"$%$:3:>+#+,':$:+*"#$)3$#;"$>"%*#$'(#"&"/$'#$:+(#%'(*5$#;"$$
:+//"*-+(0"(:"$1'#;$#;"$.%/'+?*$:+*"#*$+2$H$'*$%*$2+>>+1*$
$
$$$$$$$$$$$$$$$$$$ H8 ! C8 5$$ H H ! CI 5$$ H I ! CC 5$$ H _ ! CO 5$$ H C ! C88 5$$ H J ! C8C 6$
$
9"$"G#"(0$ H $)3$%00'(&$#;"$">","(#$ # 8D ! IPC 5$3'">0'(&$#;"$&/+?-$$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ K ! @8?H?_?T?8C?8J?HI?HO ?HU?IP> 5$
$
1;':;$'*$#;"$2?>>$&/+?-$+2$Ic/"*'0?"*6$E>#"/(%#'.">35$1"$:%($1/'#"$ K ! H 8 ; H J 6$
$
D;"$2%:#+/'[%#'+($+2$ x I8 4 8 $'(#+$'//"0?:')>"$-+>3(+,'%>*$+."/$ GF DHC $'*$$
$
$$$$ x I8 4 8 ! D x 4 8CD x C 4 x H 4 8CD x C 4 x I 4 8CD x C 4 x I 4 x H 4 x 4 8C D x C 4 x _ 4 x I 4 x H 4 8C $
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ D x C 4 x _ 4 x H 4 x 4 8CD x C 4 x _ 4 x I 4 x 4 8C 6$
$
!"#$#;"$-/','#'."$I8 th $/++#$ ) +2$?('#3$)"$0"2'("0$%*$%$["/+$+2$ x C 4 x H 4 8 6$<+5$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ) C 4 ) H 4 8 ! P 6$
$
D;"$:+(d?&%#"$/++#*$+2$ ) $%/"$ ) H 5$ ) _ 5$ ) T $%(0$ ) 8J 5$%(0$;"(:"$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ P8 D x C A! ( D x # ) l C ! x C 4 x H 4 8 6$
l$H8
$
V/+,$#;"$0"2'('(&$-+>3(+,'%>$+2$ ) 5$'#$2+>>+1*$#;%#$ ) #8 ! ) IP $'*$$%$["/+$+2$#;"$/":'-/+:%>$
-+>3(+,'%>$ x C 4 x I 4 8 6$D;"/"2+/"5$1"$;%."$*','>%/>3$%*$%)+."$#;%#$$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ PJ D x C A! ( D x # ) l C R$ x C 4 x I 4 8 5$
l$H J
$
1'#;$["/+*$ ) IP 5 ) HU 5 ) HO 5 ) HI $%(0 ) 8C 6$
$
N?)'(&$#;"$/">%#'+($ ) C 4 ) H ! 8 $-/+.'0"*$?*$1'#;$ ) 8C 4 ) 8H 4 ) U 4 ) J ! 8 5$2/+,$1;':;$
1"$'(2"/$#;%#$ ) I $'*$%$["/+$+2$#;"$-+>3(+,'%>$ x C 4 x _ 4 x I 4 x H 4 8 6$<+5$1"$:+(:>?0"$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ PH D x C A! ( D x # ) l C R$ x C 4 x _ 4 x I 4 x H 4 8 5$
l$H H
$
;%*$["/+*$ ) I 5 ) J 5 ) 8H 5 ) H_ $%(0$ ) 8O 6$$
$
$
IP
%(0$("G#5$*'(:"$ # ID ! HTC $ H _ 5$#;%#$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ P_ D x C A! ( D x # ) l C ! x C 4 x I 4 x H 4 x 4 8 5$
l$H _
$
;%*$["/+*$ ) O 5 ) 8_ 5 ) HT 5 ) HC $%(0$ ) 8U 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$" $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$ $
$
$
$
="##23&#1./&# n ! H p "#
V'/*#$1"$-/+."$#;"$2+>>+1'(&$>",,%6$
$
E&''.#4=#
If p is an odd prime and q an odd prime power with 4H p5 q 7 ! 8 , and if r @! ord p< 4q 7 ,
then ord H p< 4q7 ! r , for any integer < = P .
6-))0#
9"$;%."$#;%#$ q r ! 8 4 kp < $2+/$*+,"$'(#"&"/$ k eP5$%(0$$ r $'*$,'(',%>$1'#;$/"*-":#$#+$#;'*$
-/+-"/#36$<'(:"$p$%(0$ q $%/"$+005$ k $'*$"."(5$'6"6$ k ! Hl 6$`"(:"5$ q r ! 8 $,+0$ H p < 6$!"#$$
f
r f @! +/0 H p< 4q7 5$$#;"($ q r ! 8 4Hl f p < 5$%(0$ r f !r.$Q+15$ r f > r $1+?>0$',->3$
+/0 p< 4q7 " r f > r 6$D;'*$'*$%$:+(#/%0':#'+(5$*+$ r f ! r 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$"$$$$$$$$$$$$$$$
$ $
$
$
=($#;"$:%*"$+2$ n ! H p $1"$*;%>>$%-->3$#;"$>",,%$2+/$ < ! 8 6$
=($#;'*$:%*"5$#;"$2%:#+/'[%#'+($+2$ x n # 8 $'(#+$:3:>+#+,':$-+>3(+,'%>*$$;%*$#;"$2+/,$
$
$$$$$$$$$$$ x H p # 8 ! 4 x # 87' H 4 x7' p 4 x7' H p 4 x7 ! 4 x # 874 x 4 87' p 4 x7' p 4#x7 5$$$$$$$$$$$$$$$$$$$$$$$$4_H7$
$
1'#;$0"&$ ' p 4 x7 ! $0"&$ ' H p 4 x 7 ! p # 8 6$V+/$q 1"$#%K"$%$-/',"$-+1"/$*?:;$#;%#$ 4H p5 q 7 ! 8 6$
D;"$:3:>+#+,':$:+*"#$,+0$ H p $1'#;$/"*-":#$#+$ q $1;':;$:+//"*-+(0*$#+$ ' H 4 x7 $'*$
C p ! A pB 6$=2$ Pi 4 x7 $'*$%$,+(':$'//"0?:')>"$-+>3(+,'%>$:+(#%'("0$'($ ' p 4 x7 5$#;"($
4 #87 r Pi 4 # x 7 $'*$%$,+(':$$'//"0?:')>"$-+>3(+,'%>$:+(#%'("0$'( ' H p 4 x 7 ! ' p 4 # x 7 6$`"(:"5$
%>>$'//"0?:')>"$-+>3(+,'%>*$ Pi 4 x7 5$ i $ S SAP5 pB 5$%/"$+2$0"&/""$ r @! +/0 H p 4q 7 R$+/0 p 4q 7 5$
%(0$#;"/"$%/" H% ! H* 4H p 7 S r ! H* 4 p 7 S r $+2$*?:;$-+>3(+,'%>*6$=($&"("/%>$(+#%#'+(5$1"$
*;%>>$>%)">$#;"$:+>?,(*$4%(0$/+1*7$+2$#;"$,%#/'G$M $)3$#;"$>%)">*$$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ P5 i8 5 iH 56666665 i% 5 i% 48 5 i% 4 H 566666665 iH% 5 p $6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
4_I7$
$
I8
$
D;"$>%)">*$ i8 5 iH 566665 i% $ S $'(0':%#"$#;"$ % $,+(':$'//"0?:')>"$-+>3(+,'%>*$:+(#%'("0$'($
' H p 4 x 7 ! ' p 4 # x 7 5$1;'>"$ i% 48 5 i% 4 H 56666665 iH% $ S $'(0':%#"$#;"$,+(':$'//"0?:')>"$
-+>3(+,'%>*$:+(#%'("0$'($ ' p 4x 7 6a+/"$-/":'*">35$1"$*;%>>$:;++*"$#;"$'(0':"*$*?:;$#;%#$
Pi j 4 x7 ! 4#87 r Pi% 4 j 4# x7 6$V?/#;"/,+/"5$*'(:"$1"$*;%>>$:;++*"$%$["/+$+2$ Pi8 4 x7 $%*$-/','#'."$
H p th /++#$$+2$?('#35$1"$;%."$ i8 @! 8 6$D;"$'//"0?:')>"$-+>3(+,'%>*$ Pi 4x7 5$
1'#; i $ Ai8 5 iH 56666665 i% B 5$%/"$,'(',%>$&"("/%#+/*$+2$#;"$GRc:+0"*$ C Hi p 5q 5t 5$ t ! % 6$D;"$
'//"0?:')>"$-+>3(+,'%>*$ Pi 4x7 $1'#;$ i $ Ai% 48 5 i% 4 H 56666665 iH% B $%/"$,'(',%>$&"("/%#+/*$+2$
:+0"*$+2$>"(&#;$ H p $1;':;$$%/"$"F?'.%>"(#$#+$GRc:+0"*$ C ip 5q 5t 6$
$
E*$2+/$#;"$,%#/'G$">","(#*$+2$ M 5$#;"$">","(#*$ . P 5i 5$ i ? AP5 pB 5$%/"$"F?%>$#+$ r $%(0$$
. P5P ! . P5 p ! 8 6$V+/$#;"$+#;"/$">","(#*$+2$ M $+("$;%*$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ . j 5i ! #
mi
pij 58 ! # p ij 58 5$$ i5 j $ S SAP5 pB 5$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
mij
4__7$
1;'>"$ . j 5 P ! 8 $%(0$ . j 5 p ! . p 5 j ! 8 $2+/ j $"."($%(0$ . j 5 p ! . p 5 j ! #8 $2+/$$j$+006$=($-%/#':?>%/5$
1"$;%."$2+/$$#;"$:+>?,($'(0"G"0$)3$8$41;':;$0"#"/,'("*$#;"$'0",-+#"(#$&"("/%#+/$+2$#;"$
GRc:+0"$ CH#8p 5q 5t 75$#;"$">","(#*$ . P58 ! r 5$ . j 58 ! # p j 58 $2+/$ j ? AP5 pB $%(0$ . p 58 R #8 6$V+/$#;"$
:+>?,($'(0"G"0$)3$ #8 $41;':;$0"#"/,'("*$#;"$'0",-+#"(#$&"("/%#+/$+2$#;"$GRc
:+0" CH8 p 5q 5t 75$1"$;%."$*','>%/>3$ . P5 #8 ! r 5$ . j 5 #8 ! # p# j 58 $2+/$ j ? AP5 pB $%(0$ . p 5 #8 ! #8 6$$$$$"$
$
D;"$2+>>+1'(&$#1+$>",,%*$&'."$#;"$-/":'*"$/">%#'+(*;'-$)"#1""($#;"$:3:>+#+,':$:+*"#*$
,+0$ p %(0$#;+*"$,+0$ H p 51'#;$/"*-":#$#+$ q 5$$%(0$%>*+$1'#;$#;"$:+//"*-+(0'(&$'//"0?:')>"$
-+>3(+,'%>*$'($ GF \q ]4 x 7 6$D;"$(+#%#'+($ S f $*#%(0*$2+/$#;"$*"#$+2$'(0':"*$+2$:3:>+#+,':$
:+*"#*$,+0$ p 5$1;'>"$ S $0"(+#"*$#;"$*"#$+2$'(0':"*$+2$:3:>+#+,':$:+*"#*$,+0$ H p 6$
<','>%/>35$#;"$(+#%#'+($ C if $*#%(0*$2+/$%$:3:>+#+,':$:+*"#$,+0$ p 5$%(0$ C j $2+/$%$:3:>+#+,':$
:+*"#$,+0$ H p 6$!'K"$%>1%3*$'($#;'*$*":#'+(5$ p '*$%($+00$-/',"$%(0$ q $'*$%($+00$-/',"$
-+1"/$1'#;$ 4 p5 q 7 ! 8 5$ r @! +/0 p 4q 7 $%(0$ % @! 4 p # 87 S r 6$!"#$ Cif ! Ai5 iq5666665 iq rP #8B 5$ i $ S f 5$
rP ! r 5$)"$%$:3:>+#+,':$:+*"#$,+0$ p 6$9"$0"2'("$ CHi $#+$)"$#;"$*"#$:+(*'*#'(&$+2$%>>$
'(#"&"/*$ Ha 5$1'#;$ a $ Cif 6$$V?/#;"/,+/"5$1"$'(#/+0?:"$2+/$"%:;$ i $ S f 5$%$*"#$:+(*'*#'(&$+2$
#;"$'(#"&"/*$ a $ Cif $$1;"($$a$'*$+00$%(0$$ a 4 p $1;"($a$'*$"."(6$=2$i$'*$+005$1"$:%>>$#;'*$*"#$
Ci $%(0$'2$i '*$"."( Ci 4 p 6$=#$'*$:>"%/$#;%#$ Ci $%(0$ Ci 4 p $+(>3$:+(#%'($+00$'(#"&"/*$1;':;$%/"$
>"**$#;%($ H p 6$
$
E&''.#4F$
4'7 CHi is a cyclotomic coset mod H p for all i $ S f X$
4''7 Ci and Ci 4 p are cyclotomic cosets mod H p for all i $ S f X$
$
IH
4'''7$if $a $ Ci , then modulo 2p, Ha $ CHi X$
4'.7 the cyclotomic cosets in 4'7 together with the cyclotomic cosets in 4''7 constitute the
complete family of H% 4 H cyclotomic cosets mod H p 5$and their indices constitute the
index set SX
4.7 the cyclotomic cosets in 4'7 correspond to irreducible polynomials contained in
' p 4x 7 and those in 4''7 to irreducible polynomials contained in ' H p 4 x 7 X$
4.'7$$if Pi 4 x7 is an irreducible factor of ' p 4 x7 , then Pi 4 p 4 x7 is an irreducible factor of
' H p 4 x 7 and vice versa, for all i $ S f $and Pi 4 p 4 x7 ! 4 #87 Pi 4 # x 7 X$
4.''7$$if C j 4 p , j $ S and j odd, is the set of integers a $ C j for a even, and a 4 p for a odd,
then the family AC j 4 p @ j $ S B is the same as the family ACHi @ i $ S fB as defined in 4i7X$
4.'''7$$ CHi ! Ci 4 p 5$and hence$ PHi 4 x7 ! Pi 4 p 4 x 7 5$for all$ i $ S f 5$if and only if$$ H $ C8 6$
$
6-))0#
4'7$$D;"$'(#"&"/*$ a $%(0$ b $%/"$)+#;$">","(#*$+2$#;"$:3:>+#+,':$:>%**$ C if $'2$%(0$+(>3$'2$
b ! aq j $,+0$ p $2+/$*+,"$ j 6$=#$2+>>+1*$',,"0'%#">3$#;%#$#;'*$'*$"F?'.%>"(#$#+$ Hb ! Haq j $
,+0$ H p 5$$+/$"F?'.%>"(#>35$ a $%(0$ b $%/"$'($#;"$*%,"$:3:>+#+,':$:+*"#$,+0$ H p 6$<'(:"$%>>$
r ">","(#*$+2$ C Hi $%/"$0'22"/"(#5$#;"$*#%#","(#$(+1$2+>>+1*6$
4''7$$D%K"$#1+$'(#"&"/*$ a $%(0$ b $2/+,$ Ci f 6$=2$)+#;$%/"$+005$#;"($ b ! aq j $,+0$ p 2+/$*+,"$
j 5$+/$"F?'.%>"(#>3$ b ! aq j 4 kp 6$<'(:"$ a 5$ b 5$ q $%(0$ p %/"$+005$ k $,?*#$)"$"."($%(0$
#;"/"2+/"$ b ! a q j $,+0$ H p 6$=2$)+#;$ a $%(0$ b $%/"$"."(5$#;"$-/++2$'*$*','>%/6$Q+15$%**?,"$
a $'*$+00$%(0$ b $'*$"."(6$9"$K(+1$#;%#$ b ! aq j 4 kp $1'#;$ k $+006$`"(:"5$
b 4 p ! aq j 4 4k 4 87 p 5$%(0$*+$ b 4 p ! a q j $,+0$ H p6 $
4'''7$$D;'*$/">%#'+(*;'-$'*$',,"0'%#">3$:>"%/$2/+,$#;"$0"2'('#'+(*$+2$$ Ci $%(0$ C Hi 6$
4'.7$$<'(:"$%>>$:+*"#*$;%."$*'["$ r $%(0$*'(:"$#;"'/$'(#"/*":#'+(*$%/"$%>>$",-#35$#;"3$2+/,$%$
2%,'>3$+2$ H% $0'22"/"(#$:3:>+#+,':$:+*"#*$,+0$ H p $D+&"#;"/$1'#;$ CP $%(0$ C p 5$#;"3$2+/,$
%$:+,->"#"$2%,'>3$+2$ H% 4 H :+*"#*$,+0$ H p 6$
4.7$$E**?,"$ ) $'*$%$-/','#'."$ H p th $/++#$+2$?('#36$D;"($ ) H $'*$%$ p th $/++#$+2$?('#36$D;"$ ) c$
-+1"/*$1'#;$%($"G-+("(#$'($ C Hi $%/"$%>>$-+1"/*$+2$ ) H $%(0$#;"/"2+/"$["/+*$+2$ ' p 4x 7 6$D;"$
) c-+1"/*$1'#;$%($"G-+("(#$'($ Ci f $:%($%>>$)"$1/'##"($%*$ ) p ) H j $2+/$*+," j 6$<'(:"$
) p ! #8 5$#;"*"$-+1"/*$%/"$["/+*$+2$ ' p 4 # x 7 ! ' H p 4 x 7 6$$$$$$$$
4.'7$$D;'*$2+>>+1*$2/+,$ Pi 4 p 4 x7 !
( 4x # )
j$Ci 4 p
j
7 ! ( 4 x 4 ) j 7 ! 4#87r Pi 4# x 7 X$
j$Ci
4.''7$$D;"$2%,'>3$+2$:+*"#*$ Ci $2+/$i "."($%(0$ Ci 4 p $2+/$ i $+005$'*$-/":'*">3$#;"$2%,'>3$+2$
:3:>':$:+*"#*$:+(#%'('(&$#;"$"."($'(#"&"/*$,+0$ H p X $
4.'''7$$9"$;%."$#;"$2+>>+1'(&$"F?'.%>"(:'"*$ PHi 4 x7 ! Pi 4 p 4 x 7 @ CHi ! Ci 4 p @ Hi $ Ci 4 p $ @ $
H $ C8 .
$
$
"$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
II
$
7&'.-8#4G$
=#$,%3$+::?/$#;%#$#;"$'(0"G$ i $+2$#;"$:3:>+#+,':$:+*"#$ C i $'*$(+#$#;"$*,%>>"*#$'(#"&"/$'($#;'*$
:+*"#6$=2$*+5$1"$,%'(#%'($i$%*$'(0"G$+2$#;%#$:+*"#$$4/",",)"/$+?/$:+(."(#'+(*$+2$'(0"G'(&$$
:3:>+#+,':$:+*"#*76$Y?"$#+$#;'*$%&/"","(#$1"$;%."$#;%#$ S f $'*$%$*?)*"#$+2$#;"$:+,->"#"$*"#$
+2$'(0':"*$S5$#;+?&;$#;"$2%,'>3$+2$:+*"#*$'(0"G"0$)3$#;"$">","(#*$+2$ S f $'*$(+#$%$*?)2%,'>3$
+2$#;"$:+,->"#"$2%,'>3$+2$:+*"#*$,+0$ H p 6$
$
$
=#$2+>>+1*$2/+,$!",,%$IO$$#;%#$#;"$'//"0?:')>"$-+>3(+,'%>*$ Pi f 4 x7 5$ PiH 4 x 7 5g65$ Pi% 4 x7 $:%($
8
)"$'0"(#'2'"0$1'#;$#;"$-+>3(+,'%>*$ Pi 4 x 7 5$ i $ S fhAPB 5$%(0$#;%#$#;"$-+>3(+,'%>* Pi% 48 4 x7 5$
Pi% 4 H 4 x 7 5g665 PiH% 4 4 x7 $:+//"*-+(0$$1'#;$#;"$-+>3(+,'%>*$ Pi 4 p 4 x7 5$ i $ S fh APB 6$=#$2+>>+1*$#;%#$
#;"$:;+':"$2+/$#;"$'(0':"*$ i j 5$*?:;$#;%#$ Pi% 4 j 4 x7 ! 4#87 r Pi j 4# x7 5$2+/$ 8 " j " % 5$:%($)"$
%::+,->'*;"0$)3$#%K'(&$ i% 4 j ! i j 4 p 5$*'(:"$ ) p ! #8 $'2$ ) $'*$%$-/','#'."$ H p th $/++#$+2$?('#36$$
$E(+#;"/$$',,"0'%#"$:+(*"F?"(:"$+2$!",,%$IO$4.'7$%(0$+2$#;"$-/+-"/#3$
' H p 4 x 7 ! ' p 4 # x7 $'*$#;%#$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ pi j 58 ! # pi% 4 j 58 5$$ 8 " j " % 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$4_C7$
$
D;"$-+>3(+,'%>*$4:3:>+(+,'%>*7$ ci% 4 j 4x7 5$ 8 " j " % 5$2+/$#;"$:%*"$ n ! H p 5$:%($)"$+)#%'("0$
2/+,$#;"$$-+>3(+,'%>*$ cif j 4 x7 5$ i j $ S fh APB 5$2+/$#;"$:%*"$ n ! p 5$)3$,"%(*$+2$#;"$/">%#'+($
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ci% 4 j 4 x7 ! cif j 4 x p 48 7 5$$ 8 " j " % 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$4_J7$
=($#;"$*-":'2':$:%*"$#;%#$ H $ C8f 5$%(0$;"(:"$ H $ C84 p 5$1"$,%3$/"->%:"$ x p 48 $)3$ x H 6$
^",",)"/$#;%#$'2$$j$#%K"*$+($#;"$.%>?"*$85$H5$g5$ % 5$$#;"$'(0"G$ i j $+2$#;"$-+>3(+,'%>$
cif j 4 x7 $$/?(*$#;/+?&;$$#;"$*"#$ S fhAPB 6$V/+,$#;"$:+(*#/?:#'+($+2$#;"$:3:>+#+,':$:+*"#*$ Ci $
,+0$ H p 5$ i $ S fhAPB 5$$'#$2+>>+1*$#;%#$ ci j 4x7 $'*$+)#%'("0$2/+,$#;"$-+>3(+,'%>$ cif j 4 x7 $)3$
%00'(&$ p #+$#;"$"G-+("(#*$+2$#;"$"."($-+1"/*$+2$ x 5$%(0$>"%.'(&$'(.%/'%(#$#;"$+00$-+1"/*6$
=($%$2+/,%>$1%3$#;'*$:%($)"$"G-/"**"0$)3$#;"$/">%#'+($
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ci j 4 x7 ! x p cif j 4 x p 48 7 5$$ 8 " j " % 5$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$4_O7$
$
%*$+("$:%($."/'23$F?'#"$"%*'>35$)3$;%(0>'(&$#;"*"$-+>3(+,'%>*$,+0$ x H p # 8 6$
D;"$("G#$#;"+/",$"G-/"**"*$#;"$'0",-+#"(#*$ 1 i 4x7 $%(0$ 0i 4x7 $2+/$#;"$:%*"$ n ! H p '($
#"/,*$+2$#;"$:3:>+(+,'%>*$%(0$'0",-+#"(#*$2+/$#;"$:%*"$ n ! p 6$
$
23&)-&'#4H$
$
I_
4'7$The idempotent generator 0i 4x7 ! 8 # 1 i 4 x7 of the code C H#pi 5q 5t , i $ S & U H p , is
determined by the$expression$$
%
$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 1i 4 x7 ! 4H p7 #8\r 48 # x p 7 # , pi j i 58 4 x p # 87cif j 4 x p 48 7] X$
j !8
4''7$If 1 4 x 7 is the idempotent corresponding to column i of the matrix M f for n ! p ,
then
1 i 4 x 7 ! #H #8 4 x p # 871 if 4 x p 48 7 X$
$
4'''7$$If 0if 4 x 7 is the idempotent generator of the GR-code C p#5iq 5t , then
f
i
0i 4 x 7 ! H #8\8 4 x p # 4 x p # 870if 4 x p 48 7] .
$
6-))0$
D;"$"G-/"**'+(*$2+>>+1$',,"0'%#">3$2/+,$D;"+/",$IH$%(0$2/+,$#;"$/">%#'+(*$4_C75$4_J7$
%(0$4_O76$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$"$$$$$
$
$
:;.'(<&#!I#
D%K"$ n ! 8P %(0$ q ! I 6$=#$2+>>+1*$#;%#$ r ! _ 5$ % ! * 48P7 S _ ! 8 6$D;"$:3:>+#+,':$:+*"#*$
,+0$8P$%/"$ CP ! APB 5$ C8 ! A85I5 O5UB 5$ CH ! AH5 _5 J5TB 5$ CC ! ACB 5$%(0$#;"$:+//"*-+(0'(&$
-+>3(+,'%>*$%/"$ PP 4 x7 ! x # 8 5$ P8 4 x 7 ! x _ # x I 4 x H # x 4 8 5$ PH 4 x7 ! x _ 4 x I 4 x H 4 x 4 8 5$
PC 4 x7 ! x 4 8 6$$
D;"$,%#/'G$ M $'*$"F?%>$#+$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$P$$$$$8$$$$$H$$$$$$C$
58 8 8 8 6 P
7
8
8 8 #8 #88 8
7
6$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ M !
78 #8 #8 8 8 H
7
8
98 #8 8 #8: C
$
8
<'(:"$ t ! % ! 8 5$#;"/"$'*$+(>3$+("$&"("/%>'["0$/"*'0?"$:+0"$ C8P5I58
5$&"("/%#"0$)3$'#*$
,'(',%>$-+>3(+,'%>$ P8 4 x7 6$`"/"5$#;"$:3:>+#+,':$:+*"#*$%/"$*">2c'(."/*"5$%(0$#;"/"2+/"$#;"$
8
'0",-+#"(#$&"("/%#+/$+2$#;'*$:+0"$ C8P5O58
$'*$0"#"/,'("0$)3$#;"$:+>?,($1'#;$>%)">$86$V'/*#$
1"$2'(0$$
$
$$$ 18 4 x 7 ! 8P #8 4cP 4 x 7 4 c8 4 x7 # cH 4 x 7 # cC 4 x7 7R$$8i x 4 x I 4 x O 4 x U # x H # x _ # x J # xT # x C 5$
$
%(0$("G#$$
$
$
IC
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 08 4 x7 ! 8 # 18 4 x 7 ! # x U 4 xT # x O 4 x J 4 x C 4 x _ # x I 4 x H # x 6$$$$$$$$$$$$$$$
$
Q+15$1"$*;%>>$:+,-?#"$#;"*"$'0",-+#"(#*$)3$%-->3'(&$-%/#*$4''7$%(0$4'''7$+2$D;"+/",$IU6$
D;"$:3:>+#+,':$:+*"#*$%/"$ C P APB 5$ C8 ! A85H5I5_B 6$D;"$GRc:+0"$ C C85I58 $'*$&"("/%#"0$)3$$
P8 4 x7 ! x _ 4 x I 4 x H 4 x 4 8 6$$$6$
D;"$,%#/'G$ M f $'*$%$ H A H c*?),%#/'G$+2$ M 5$1;':;$'*$+)#%'("0$)3$#%K'(&$#;"$2'/*#$#1+$
:+>?,(*$%(0$#;"$2'/*#$%(0$#;'/0$/+1$+2$ M $
$
58 8 6
88 6$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ M f ! 77
9 8 # 8:
$
V/+,$D;"+/",$IH$'#$2+>>+1*$*#/%'&;#2+/1%/0>3$#;%#$ 18f 4 x7 ! #8 4 x 4 x H 4 x I 4 x _ 5$%(0$
;"(:"$ 08f 4 x 7 ! # x _ # x I # x H # x # 8 6$9;"($*?)*#'#?#'(&$#;"*"$"G-/"**'+(*$'($#;"$2+/,?>%*$
+2$D;"+/",$IU$4''7$%(0$4'''7$/"*-":#'.">35$%(0$?*'(&$#;%#$ H #8 ! #8 $'($ GF \I] 5$+("$+)#%'(*$
#;"$*%,"$/"*?>#*$2+/$ 18 4 x7 $%(0$ 08 4 x7 %*$)"2+/"6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$"$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$$$$$$$$$$$$$$$$$$$$
$
F"##23&#1./&# n ! p < 6$
Q"G#5$1"$#%K"$ n ! p < $%(0$ q $%$-/',"$-+1"/$*?:;$#;%#$ 4 p5 q 7 ! 8 6$
=($+/0"/$#+$0"%>$1'#;$#;'*$:%*"5$1"$(""0$#;"$2+>>+1'(&$>",,%6$
$
E&''.#!J$
Let p be a prime and q be some prime power with 4 p5 q 7 ! 8 . Let furthermore a be some
positive integer. For p odd and a B 8 or for p even and a B H , the inequality
ord p a 48 4q 7 = ord p a 4q 7 implies $ord p a 4 i #8 4q 7 ! p i #8ord p a 4q 7 5$for all$$ i B 8 6$
6-))0#
4'7$V'/*#$1"$:+(*'0"/$#;"$:%*"$ p +006$V+/$+?/$:+(."('"(:"5$1"$%**?,"$ a ! 8 5$%(0$*+$#;"$
:+(0'#'+($+2$#;"$!",,%$'*$+/0 pH 4q7 = +/0 p 4q 7 6$
!"#$ r @! +/0 p 4q 7 6$9"$*;%>>$-/+."$)3$,%#;",%#':%>$'(0?:#'+($#;"$2+>>+1'(&$*"#$+2$/">%#'+(*@$
$
i #8
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$+/0 p i 4 q 7 ! rp i #8 5$$ q rp ! 8 4 k i p i 5$$ 4k i 5 p7 ! 8 5$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$4_T7$
$
2+/$%>>$ i B 8 6$V/+,$#;"$:+(0'#'+($+/0 p H 4 q 7 = $+/0 p 4q 7 5$'#$2+>>+1*$#;%#$1"$:%($1/'#"$
q r ! 8 4 kp 5$1'#;$ 4 k 5 p 7 ! 8 6$<+5$4_T7$;+>0*$2+/$ i ! 8 5$)3$0"2'('#'+($+2$ r $%(0$)3$#%K'(&$
k8 ! k 6$
Q"G#5$1"$%**?,"$#;%#$$#;"$/">%#'+(*$4_T7$%/"$#/?"$2+/$*+,"$2'G"0$'(#"&"/$ i B 8 6$D;"($
q rp
$
$
i #8
! 8 4 k i p i 5$2+/$*+,"$'(#"&"/$ k i $1'#;$ 4k i 5 p 7 ! 8 6$=#$2+>>+1*$#;%#$$
IJ
p
i
5 p6
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ q rp ! 48 4 k i p i 7 p ! , 77 884k i p i 7 l $
l !P 9 l :
p #8
5 p6
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$R$ 8 4 k i p i 48 4 , 77 884k i p i 7 l 4 k i p pi 6$
l !H 9 l :
$
5 p6
<'(:"$$p! 7 8 $2+/$%>>$ l $ AH5I5666665 p # 8B 5$%>>$#"/,*$'($#;"$*?,,%(0$%/"$0'.'*')>"$)3$ p i 4 H 6$
9l:
a+/"+."/5$*'(:"$ p '*$+005$1"$;%."$ pi B Ii B i 4 H 5$%(0$*+$#;"$>%*#$#"/,$'($#;"$/;*$'*$%>*+$
i
0'.'*')>"$)3$ p i 4 H 6$D;"/"2+/"5$1"$:%($1/'#"$ q rp ! 8 4 k i 48 p i 48 $1'#;$ 4k i 48 5 p7 ! 8 5$+/$
"F?'.%>"(#>35$+/0 p i 48 4q 7 ! rp i 6$$<+5$'2$ e @! +/0 p i 48 4q 7 5$1"$;%."$ rp i ! ey $2+/$*+,"$'(#"&"/$y. $
9"$%>*+$K(+1$#;%#$ e ! xrp i #8 $2+/$*+,"$'(#"&"/ x 6$N+,)'('(&$#;"*"$#1+$"F?%>'#'"*$&'."*$
xy ! p 5$*+$"'#;"/$ x ! 8 $+/$ x ! p 6$Z?#$ x ! 8 $3'">0*$$+/0 pi 48 4q 7 ! +/0 p i 4q 7 ! rp i #8 5$1;':;$
:+(#/%0':#*$4_T76$`"(:"5$+/0 p i 48 4 q 7 ! rp i 6$$$$$<+5$#;"$/">%#'+(*$4_T7$$%>*+$;+>0$2+/$ i 4 8 6$Y?"$
#+$#;"$-/'(:'->"$+2$,%#;",%#':%>$'(0?:#'+(5$#;"$/">%#'+(*$4_T7$;+>0$2+/$%>>$ i B 8 6$D;"$
"F?%>'#3$+2$#;"$!",,%$(+1$2+>>+1*$',,"0'%#">36$
V+/$.%>?"*$ a = 8 #;"$-/++2$'*$:+,->"#">3$*','>%/6$
$
4''7$Q"G#5$1"$:+(*'0"/$#;"$:%*"$ p ! H 5$%(0$1"$#%K"$ a ! H 6$
!"#$ r f@! +/0 H H 4 q 7 6$D;"$:+(0'#'+($+2$#;"$!",,%$3'">0*$+/0 HI 4q 7 = $+/0 H H 4 q 7 6$9"$*;%>>$
-/+."$)3$'(0?:#'+($#;%#$2+/$%>>$ i B 8 $$$$$$
$
i #8
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$+/0 Hi 48 4q7 ! r fH i #8 5$$ q r fH ! 8 4 k i H i 48 5$$ 4 k i 5H7 ! 8 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
4_U76$
$
V+/$ i ! 8 $#;"*"$/">%#'+(*$%/"$#/?"5$)":%?*"$+2$#;"$0"2'('#'+($+2$ r f 6$$$$
i #8
E**?,"$4_U7$;+>0*$2+/$%$:"/#%'($.%>?"$+2$ i B 8 $6$D;"($$ q r fH ! 8 4 k i H i 48 $1'#;$ 4k i 5H7 ! 8 6$
=#$2+>>+1*$#;%#$$
$
i
H
$$$$$$$$$$$$$$$ q r fH ! 48 4 k i H i 48 7 H ! 8 4 k i H i 4 H 4 k i H Hi 4 H ! 8 4 k i 48 H i 4 H 5$$$ 4k i 48 5H7 ! 8 6$$
$
<'(:"$ i B 8 5$#;'*$',->'"*$+/0 Hi 4 H 4 q 7 ! r fH i 6$D;"/"2+/"5$'2$ ef@! $+/0 Hi 4 H 4 q 7 5$1"$:%($1/'#"$$
r fH i ! ef y 6$9"$%>*+$;%."$ ef ! xr fH i #8 6$`"(:"5$ xy ! H 5$%(0$*+$ x ! 8 $+/$ x ! H 6$Z?#$ x ! 8 $
&'."*$+/0 Hi 4 H 4q7 ! r fH i #8 ! $+/0 Hi 48 4 q 7 $1;':;$:+(#/%0':#*$4_U76$<+$ x ! H $%(0$;"(:"$4_U7$
%>*+$;+>0*$2+/$ i 4 8 5$%(0$)3$'(0?:#'+($2+/$%>>$ i B 8 6$
V+/$.%>?"*$ a = H $#;"$-/++2$'*$:+,->"#">3$*','>%/6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
"$
$
$
$
IO
$
$
7&'.-8#!5$
D;"$%)+."$/"*?>#$:%($%>*+$)"$0"/'."0$2/+,$\H5$!",,%$I6I_]6$E:#?%>>35$!",,%$_8$'*$%$
*-":'%>$:%*"$+2$#;"$>",,%$'($\H]6$
$
7&'.-8#!4##
D;"$:%*"$ p ! H 5$ q ! I $*;+1*$#;%#$#;"$:+(0'#'+($ a B H $'*$(":"**%/35$*'(:"$+/0 HH 4I7 ! H = $$
+/0 H8 4I7 ! 8 5$)?#$+/0 HI 4I7 ! H 3 _ 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$
$
E*$?*?%>$1"$;%."$ r @! +/0 n 4q 7 ! +/0 p< 4q 7 6$a+/"+."/5$1"$0"2'("$u@R+/0 p 4q 7 6$=($#;'*$:%*"$
#;"$2%:#+/'[%#'+($+2$ x n # 8 $'(#+$:3:>+#+,':$-+>3(+,'%>*$'*$%*$2+>>+1*$
$
<
$$$$$$$$$$$$$$$$$$$$$$$$$ x n # 8 ! x p # 8 ! 4 x # 87' p 4 x7' p H 4 x76666666' p< 4 x7 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$4CP7$
$
D;"$:3:>+#+,':$-+>3(+,'%>$ ' p 4x 7 $'*$#;"$-/+0?:#$+2$C @! * 4 p 7 S u ! 4 p # 87 S u $,+(':$
-+>3(+,'%>*$+2$0"&/""$ u $1;':;$%/"$'//"0?:')>"$+."/$ GF 4q 7 5$*%3$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ' p 4 x 7 ! P8 4 x 7 PH 4 x 766666666PC 4 x 7 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$4C87$
$
<','>%/>35$ ' p b 4x 7 $'*$#;"$-/+0?:#$+2$ % b @! * 4 p b 7 S rb ! p b #8 4 p # 87 S rb $'//"0?:')>"$
-+>3(+,'%>*$$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ' p b 4 x7 ! F8 4 x7 FH 4 x766666666F% b 4 x7 5$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$4CH7$
$
1;':;$%/"$%>>$+2$0"&/""$ rb 5$1;'>"$ rb ! $+/0 p b 4q 7 5$ r8 ! u $%(0$ r< ! r 6$
9"$(+1$0'*#'(&?'*;$)"#1""($p '*$+00$%(0$p$R$H6$=2$p '*$+00$%(0$'2$+/0 pH 4q7 = $+/0 p 4q 7 5$'#$
2+>>+1*$2/+,$!",,%$_8$4'7$#;%#$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ rb ! up b #8 5$ 8 " b " < 5$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$4CI7$
%(0$;"(:"$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ r S u ! n S p 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$4C_7$
$
E*$%$:+(*"F?"(:"$1"$;%."$#;%#$4:26$\$85$<":#'+($_]7$ % ! % b ! C 5$%(0$,+/"+."/$#;%#$#;"$
-+>3(+,'%>*$ Fi 4x7 $:%($)"$'0"(#'2'"0$1'#;$ Pi 4 x n S p 7 5$2+/$ 8 " i " % 6$`"(:"5$#;"$:3:>+#+,':$
-+>3(+,'%>*$'($4CH7$:%($)"$2%:#+/'["0$'(#+$'//"0?:')>"$-+>3(+,'%>*$+."/$ GF 4q 7 $%*$
2+>>+1*$
$
b #8
b #8
b #8
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ' p b 4 x7 ! P8 4 x p 7 PH 4 x p 7666666666P% 4 x p 7 5$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$4CC7$
$
IT
$
2+/$%>>$ b 5$ 8 " b " < 6$N;%(&'(&$#;"$(+#%#'+($+2$#;"$>%)">*$%*$'(#/+0?:"0$'($4H75$1"$(+1$
1/'#"$
$$
b #8
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ' p b 4 x7 ! ( Pi 4 x p 7 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
i$S fhAPB
4CJ7$$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
=($"F6$4CJ7$$i /?(*$#;/+?&;$#;"$ t 4 ! % 7 $0'22"/"(#$.%>?"*$+2$#;"$*"#$ S fhAPB R$ Ai8 5 iH 5666665 i% B 5$
1;"/"$ S f $*#%(0*$2+/$#;"$*"#$+2$'(0':"*$+2$#;"$:3:>+#+,':$:+*"#*$,+0$ p 6$D;'*$(+#%#'+($$
"G-/"**"*$#;"$+("c#+c+("$/">%#'+(*;'-$)"#1""($#;"$:3:>+#+,':$:+*"#*$,+0$p 1'#;$/"*-":#$
b #8
#+$q5 %(0$#;"$'//"0?:')>"$2%:#+/*$+2$ ' p b 4x 7 $'($ GF 4q 7\ x] 6$D;"$-+>3(+,'%>*$ Pi 4 x p 7 $%/"$$
< #8
%>>$+2$0"&/""$ rb $$2+/$%$2'G"0$.%>?"$+2$b6$=($-%/#':?>%/5$#;"$-+>3(+,'%>*$ Pi 4 x p 7 5$ i $ S fhAPB 5$
1;':;$%/"$+2$0"&/""$ r< ! r 5$%/"$#;"$,'(',%>$&"("/%#+/*$+2$#;"$&"("/%>'["0$/"*'0?"$:+0"*$
C ip< 5q 5t 5$1'#;$ t ! % 6$
$D;"*"$:+(*'0"/%#'+(*$%>*+$;+>0$2+/$ p ! H 5$'2$1"$/"F?'/"$+/0 HI 4q 7 = +/0 HH 4q 7 = +/0 H 4q7 $4:26$
!",,%$_876$D;"/"2+/"5$2/+,$(+1$+($1"$%**?,"$#;%#$+/0 p H 4 q 7 = ord p 4q 7 $%(0$
%00'#'+(%>>35$'2$ p ! H $#;%#$+/0 HI 4q 7 = +/0 H H 4 q 7 6$9"$*;%>>$0"(+#"$#;'*$:+(0'#'+($)3$4W76$$
< #8
D;"$:+>?,(*$+2$#;"$,%#/'G$ M $1;':;$:+//"*-+(0$#+$#;"$-+>3(+,'%>*$ Pi 4 x p 7 5$ i $ S fhAPB 5$
0"#"/,'("$#;"$'0",-+#"(#$&"("/%#+/*$+2$#;"$GRc:+0"*$ C ip< 5q 5t 6$=($&"("/%>5$1"$*;%>>$>%)">$
#;"$:+>?,(*$+2$ M $)3$
$
$$$$$$$$$$$$$$$$$$$$$$$$$ P5 i8< 5 iH< 56666665 i%< 5 i8< #8 5 iH< #8 56666665 i%< #8 5666666666665 i88 5 iH8 56666665 i%8 5$$$$$$$$$$$$$$$$$$$$$$$$$$4CO7$
$
1'#;$ i8< ! 8 5$%(0$*?:;$#;%#$#;"$'(0':"*$ i8b 5 iHb 56666665 i%b $$'(0':%#"$#;"$ % '//"0?:')>"$-+>3(+,'%>*$
b #8
Pi 4 x p 7 5$ i $ S fhAPB $5$2+/$ 8 " b " < 6$D;"$'(0':"*$'($4CO7$$2+/,$#;"$:+,->"#"$*"#$ S $+2$
'(0':"*$+2$#;"$:3:>+#+,':$:+*"#*$,+0$ p < 6$
D;"$2+>>+1'(&$>",,%$1'>>$"(%)>"$?*$#+$"*#%)>'*;$#;"$+("c#+c+("$/">%#'+(*;'-$)"#1""($#;"$
<
'//"0?:')>"$-+>3(+,'%>*$+."/$ GF 4q 7 1;':;$%/"$:+(#%'("0$'($ x p # 8 $%(0$#;"$:3:>+#+,':$
:+*"#*$$,+0$ p < $1'#;$/"*-":#$#+$q.$
$
E&''.#!!#
Let condition 4W7$hold. If C i ! Ai5 iq 56666665 iq rb #8 B is a cyclotomic coset mod p b with respect
to q and of size rb , then pC i ! A pi5 piq 56666665 piq rb #8 B is a cyclotomic coset mod p b 48 , also
of size rb , while it is a cyclotomic coset mod p b of size rb S p , for 8 " b " < , if one
replaces integers which occur more than once by just one of these. #
$
IU
D;"$-/++2$'*$*#/%'&;#2+/1%/0$%(0$'*$+,'##"06$E::+/0'(&$#+$#;"$/?>"$%&/""0$?-+($'($<":#'+($
I5$1"$0"(+#"$#;'*$:3:>+#+,':$:+*"#$,+0$ p b $)3$ C ip $4:26$"F6$4HJ7$%(0$!",,%$HP76$Q+15$
1"$:+(*'0"/$#;"$:3:>+#+,':$:+*"#*$'($4CO7$'(0"G"0$)3$ i8< 5 iH< 566666665 i%< 6$<'(:"$#;"*"$%/"$
:+*"#*$,+0$ p < 5$'#$2+>>+1*$2/+,$$!",,%$__$$#;%#$ i <j p ! i <j #8 5$2+/$ 8 " j " % 6$
a+/"$&"("/%>>35$'#$2+>>+1*$2+/$ 8 " b " < $#;%#$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ i <j pb #8 ! i <j #b 48 5$$ 8 " j " % 5$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$4CT7$$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ i <j ! i j 5$$ 8 " j " % 5$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$4CU7$
$
1;"/"$#;"$'(#"&"/*$ i j 5$ 8 " j " % 5$%/"$#;"$(+(c["/+$'(0':"*$+2$#;"$:3:>+#+,':$:+*"#*$,+0$$p5$
'6"6$#;"$">","(#*$+2$ S fhAPB 6$
D;"$:3:>+#+,':$:+*"#*$1'#;$'(0':"*$ i8b 5 iHb 5666665 i%b $$%/"$+2$*'["$ r S p < #b ! rb 5$$%(0$*+$#;"*"$
b #8
:+*"#*$:+//"*-+(0$#+$#;"$'//"0?:')>"$-+>3(+,'%>*$ Pi 4 x p 7 5$ 8 " i " % 5$1;':;$;%."$0"&/""$
rb 5$2+/$%>>$b5$ 8 " b " < 6$$=($-%/#':?>%/$1"$,%3$%**?,"$#;%#$#;"$:3:>+#+,':$:+*"#$ C i $,+0$
< #8
p < $:+//"*-+(0*$#+$#;"$-+>3(+,'%>$ Pi 4 x p 7 $2+/$%>>$ i $ Ai8< 5 iH< 56666665 i%< B 6$
$
Q"G#5$1"$:+(*'0"/$#;"$,%#/'G$">","(#*$ . j5i 5$ i $ Ai8< 5 iH< 566666665 i%< B 6$9"$K(+1$%>/"%03$#;%#$$
. P5i ! r $2+/$#;"*"$ i c.%>?"*6$V?/#;"/,+/"5$ p ij 58 ! P $'2$ ij ? Ai88 5 iH8 566666665 i%8 B 5$+/$*'(:"$
i $ Ai8< 5 iH< 566665 i%< B 5$ pij 58 ! P $2+/$%>>$ j ? Ai88 5 iH8 5666665 i%8 B 6$`"(:"5$1"$+)#%'($2/+,$4IO7$#;%#$
. j 5i ! P $2+/$%>>$#;"*"$'(0':"*6$<+5$1"$+(>3$;%."$#+$2+:?*$+($">","(#*$ . j5i $1'#;$
j $ Ai88 5 iH8 56666665 i%8 B 6$9"$*;%>>$/"*#/':#$+?/*">."*$#+$ i ! i8< R876$9"$/",'(0$#;"$/"%0"/$+2$#;"$
2%:#$#;%#$#;"$:+>?,(*$>%)">"0$)3$ iH< 566666665 i%< $%/"$-"/,?#%#'+(*$+2$$:+>?,($ i8< 6$
$
9"$%/"$(+1$/"%03$#+$-/+."$+?/$("G#$#;"+/",6$=($#;%#$#;"+/",5$#;"$*"#$ S a $'*$#;"$*"#$+2$#;"$
'(0':"*$+2$#;"$:3:>+#+,':$:+*"#*$,+0$ p a 6$$<+5$ S < $'*$'0"(#':%>$1'#;$ S $%(0$ S 8 $1'#;$ S f 6$
V+/$ a " < $+("$:%($"%*'>3$."/'23$#;%#$ S a $$'*$%$*?)*"#$+2$ S 4 ! S < 7 $%(0$#;%#$ S a & U pa $'*$%$
*?)*"#$+2$ S & U p< 6$9"$",-;%*'["$#;%#$#;/+?&;+?#$#;"$#"G#$+2$#;'*$*":#'+($1"$%**?,"$#;%#$
:+(0'#'+($4W7$;+>0*6$
$
23&)-&'#!9#
Let a be some integer with 8 " a " < for $p odd, and H " a " < for p even. Let
i $ S a & U pa . Let 0iW 4 x7 , i $ S & U p< , be the idempotent generator of the code C ip< 5q 5t ,
and let 0iaW 4 x7 be the idempotent generator of the code C ip a 5q 5t , i $ S a & U pa . If
< #a
ord p a 48 4q 7 = ord p a 4q 7 , then 0iW 4 x 7 ! 0iaW 4 x p 7 .
$
_P
6-))0#
V+/$+?/$:+(."('"(:"$1"$#%K"$ p $+00$#+&"#;"/$1'#;$ a ! 8 6$
V/+,$D;"+/",$IH$1"$K(+1$#;%#$ 0i 4 x7 ! 8 # 1 i 4 x7 $1'#;$ 1 i 4 x7 ! n #8 , #
k$S
r
pik 58c k 4 x7 5$
mik
<
n ! p 6$E&%'($2+/$/"%*+(*$+2$:+(."('"(:"5$1"$%**?,"$ i ! 8 6$9"$%>/"%03$%/&?"0$#;%#$#;"$
+(>3$ k c.%>?"*$#;%#$:+(#/')?#"$'($#;"$"G-/"**'+($2+/$ 1 i 4x7 %/"$ k ! P $%(0$ k $ Ai88 5 iH8 56666665 i%8 B 6$
r
V/+,$ ik8 ! ik p < #8 $'#$2+>>+1*$#;%#$ mi8 ! mik 5$%(0$;"(:"5$ . i8 58 ! #
pi 58 6$Z":%?*"$+2$#;"$
k
k
mi k k
r n
! ! p < #8 5$%(0$*+$$$$$
u p
u
r
pi 58 ! p #8 . fik 58 6$$
pik 58 ! #
!#
pmik k
nmik
:+(0'#'+($'($#;"$D;"+/",5$1"$;%."$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ n #8 . i8 58
k
`"/"5$ . fik 58 $'*$%($">","(#$+2$#;"$,%#/'G$ M f $2+/$#;"$:%*"$#;%#$#;"$:+0"$>"(&#;$'*$"F?%>$#+$ p 6$$
< #8
<'(:"$ ci8 4 x7 ! cik 4 x p 7 5$#;"$*#%#","(#$'($#;"$D;"+/",$(+1$2+>>+1*$',,"0'%#">3$2+/$#;'*$
k
:%*"6$V+/$+#;"/$a-.%>?"*$1"$%-->3$!",,%$_85$#+$+)#%'($%$:+,->"#">3$*','>%/$-/++26$$$$$$$$$"$
$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$
$
$
#
7&'.-8#!=#
< #8
D;"$%)+."$/"*?>#$:%($%>*+$)"$0"/'."0$2/+,$#;"$2%:#$#;%#$ Pi 4 x p 7 $'*$#;"$,'(',%>$&"("/%#+/$
+2$ C ip a4< #8 5 q 5t 5$'2 Pi 4x7 $'*$#;"$,'(',%>$&"("/%#+/$+2$ C ip a 5q 5t 6$
$
$
:;.'(<&#!F#
9"$#%K"$ n ! HO $%(0$ q ! O 6$<+5$+/0 I 4O7 ! 8 > $+/0 U 4O7 ! I 6$<'(:"$ u ! 8 5$'#$2+>>+1*$#;%#$
r @! +/0 II 4O7 ! U 6$D;"$:3:>+#+,':$:+*"#*$,+0$HO$%/"$$
C P ! APB 5$ C8 ! A85O5HH58U5HC58I58P58J5_B 5$ C H ! AH58_58O5885HI5HJ58U5C5TB 5$
C I ! AI5H858HB 5 C J ! AJ58C5H_B 5$ C U ! AUB $%(0$ C8T ! A8TB 6$
9"$:+(:>?0"$#;%#$ ' HO 4 x7 $'*$#;"$-/+0?:#$+2$#1+$'//"0?:')>"$-+>3(+,'%>*$+."/$ GF 4O7 $+2$
0"&/""$U5$ ' U 4 x7 '*$#;"$-/+0?:#$+2$#1+$'//"0?:')>"$-+>3(+,'%>*$+2$0"&/""$I$%(0$ ' I 4 x7 $'*$
#;"$-/+0?:#$+2$#1+$'//"0?:')>"$-+>3(+,'%>*$+2$0"&/""$8$%>>$+."/ GF 4O7 6$E:#?%>>35$1"$;%."$
$
$ x HO # 8 ! 4 x # 87' I 4 x 7' U 4 x 7' HO 4 x 7 ! 4 x # 874 x # H74 x # _74 x I # H74 x I # _74 x U # H74 x U # _7 6$
$
D;'*$'*$'($%::+/0%(:"$1'#;$#;"$2+/,%>'*,$+($-6$IT5$*'(:"$+/0 U 4O7 ! I = +/0 I 4O7 R86$
j("$:%($"%*'>3$."/'23$#;%#$#;"$'//"0?:')>"$2%:#+/*$'($#;"$/;*$:+//"*-+(0$$#+$#;"$:3:>+#+,':$
:+*"#*$)3$,"%(*$+2$#;"$'(0':"*$'($#;"$2+>>+1'(&$1%3@$
$
_8
$
PP 4 x7 ! x # 8 5$ P8 4 x7 ! x U # H 5$ PH 4 x7 ! x U # _ 5$ PI 4 x 7 ! x I # H 5$ PJ 4 x7 ! x I # _ 5$
PU 4 x7 ! x # H $%(0$ P8T 4 x7 ! x # _ 6$
$
k*'(&$#;"*"$2%:#*5$1"$0"/'."$#;"$,%#/'G$$">","(#*$+2$#;"$:+>?,($1'#;$'(0"G$8@$
$
m
U
. P 58 ! m8 ! U 5$ . P5 H ! m H ! U 5$ . U58 ! # 8 pU58 ! # 4#H7 ! 8T 5$
8
mU
m
U
.8T58 ! # 8 p8T58 ! # 4#_7 ! IJ 5$1;'>"$%>>$+#;"/$">","(#*$ . j 58 $'($#;'*$:+>?,($.%('*;5$
8
m8T
)":%?*"$ p j 58 ! P $2+/$#;+*"$">","(#*6$E2#"/$;%.'(&$:+,-?#"0$#;"$">","(#*$+2$#;"$+#;"/$
:+>?,(*$'($%$:+,->"#">3$*','>%/$1%3$%(0$/"0?:'(&$,+0$O5$1"$$2'(%>>3$2'(0$2+/$ M $
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$P$$$$8$$$$H$$$$I$$$J$$$$U$$8T$
58 U U I I 8 8 6 P
8
7
78 P P P P H _ 8 8
78 P P P P _ H 8 H
8
7
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ M ! 78 P P J C 8 8 8 $ I 6$
78 P P C J 8 8 8 J
8
7
78 _ 8 I I 8 8 8 U
8
7
98 8 _ I I 8 8 : 8T
$
V/+,$#;'*$,%#/'G$1"$0"/'."$#;"$-/','#'."$'0",-+#"(#$&"("/%#+/$
$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 18 4 x7 ! HO #8 4 . P5 H cP 4 x7 4 . U5 H cU 4 x7 4 .8T5 H c8T5 H 4 x7 7$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$R # 4HcP 4 x7 4 cU 4 x7 4 _c8T 4 x77 $
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$R I x 8T 4 J x U 4 C 6$
$
H
8
%(0$*','>%/>3$ 1 H 4 x7 ! J x 8T 4 I x U 4 C 6$D;"$'0",-+#"(#$&"("/%#+/*$+2$ C HO
5 O 5 H $%(0$ C HO5 O 5 H $%/"$
08 4 x7 ! _ x8T 4 x U 4 I $%(0$ 08W 4 x7 R$ 0H 4 x7 ! x8T 4 _ x U 4 I $%(0 0HW 4 x 7 ! 08 4 x7 ! _ x8T 4 x U 4 I 5$
/"*-":#'.">36$E*$+("$:%($."/'235$#;"*"$&"("/%#+/*$:%($)"$+)#%'("0$2/+,$#;"$:+//"*-+(0'(&$
&"("/%#+/*$+2$#;"$:+0"*$ CUi 5O5 H $)3$*?)*#'#?#'(&$ x U $2+/ x 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$"$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$
:;.'(<&#!G$
V+/$ n ! H I $%(0$ q ! I 5$1"$;%."$ r @! +/0 HI 4I7 ! H $%(0$ t ! % ! * 4T7 S H ! H 6$D;"$:3:>+#+,':$
:+*"#*$,+0$ H I $%/"$ C P ! APB 5$ C8 ! A85IB 5$ C H ! AH5JB 5 C _ ! A_B 5 C C ! AC5OB 5$',->3'(&$
S ! AP585H5_5CB 6$D;"$:+//"*-+(0'(&$'//"0?:')>"$-+>3(+,'%>*$%/"$/"*-":#'.">3$ PP 4 x7 ! x # 8 5$
$
_H
P8 4 x7 ! x H # x # 8 5$ PH 4 x7 ! x H 4 8 5$ P_ 4 x7 ! x 4 8 $%(0$ PC 4 x7 ! x H 4 x 4 8 6$D;"$2%:#+/'[%#'+($
+2$ x T # 8 $'(#+$:3:>+#+,':$-+>3(+,'%>*$'*$ x T # 8 ! ' 8 4 x 7' H 4 x 7' _ 4 x7' T 4 x 7 5$1'#;$
' 8 4 x7 ! PP 4 x7 5$ ' H 4 x7 ! P_ 4 x7 5$ ' _ 4 x7 ! PH 4 x7 $%(0$ ' T 4 x7 ! P8 4 x7 PC 4 x7 6$$$
E*$+("$:%($*""5$#;"$(?,)"/$+2$'//"0?:')>"$-+>3(+,'%>*$:+(#%'("0$'($ ' _ 4 x7 $4%(0$'($
' H 4 x7 7$'*$>"**$#;%($#;"$(?,)"/$+2$'//"0?:')>"$2%:#+/*$+2$ ' T 4 x7 6$D;'*$'*$0?"$#+$#;"$2%:#$
#;%#$#;"$:+(0'#'+($+/0 HI 4I7 = +/0 HH 4I7 $'*$(+#$*%#'*2'"05$*'(:"$)+#;$%/"$"F?%>$#+$H6$$
D;"$'//"0?:')>"$-+>3(+,'%>*$ P8 4 x7 $%(0$ PC 4 x7 $%/"$#;"$,'(',%>$&"("/%#+/*$+2$#;"$#1+$
"F?'.%>"(#$GRc:+0"*$ CT85I5 H $%(0$ CTC5I5 H 6$E-->3'(&$$#;"$"G-/"**'+($4HO7$2+/$#;"$,%#/'G$
">","(#*$ . j5i 5$ i5 j $ S 5$1"$2'(0$2+/$#;"$,%#/'G$ M $$
$
$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$P$$$$$8$$$$$C$$$$$$$H$$$$$$$_$
58 # 8 # 8 # 8 8 6 P
7
8
7 8 8 # 8 P # 88 8
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ M ! 78 # 8 8
P # 88 C 6$
7
8
88H
78 P P 8
78 8
8 # 8 8 8: _
9
$
D;"$-/','#'."$'0",-+#"(#$&"("/%#+/*$:+//"*-+(0'(&$#+$#;"$:+>?,(*$1'#;$>%)">*$8$%(0$C$
%/"$/"*-":#'.">3$$
$
$$$$$$$$$$$$$$$$$$ 1 8 4 x 7 ! T #8 4 #c P 4 x 7 4 c8 4 x 7 # cC 4 x 7 4 c _ 4 x 77 ! x O 4 x C # x _ # x I # x 4 8 5$
$
$$$$$$$$$$$$$$$$$$ 1 C 4 x7 ! T #8 4#c P 4 x7 # c8 4 x7 4 cC 4 x 7 4 c _ 4 x 77 ! # x O # x C # x _ # x I # x 4 8 6$
$
D;"$'0",-+#"(#$&"("/%#+/*$+2$ CT85I5 H $%(0$ CTC5I5 H $%/"$/"*-":#'.">3$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$ 08W 4 x 7 ! 0C 4 x 7 ! 8 # 1 C 4 x 7 ! x O 4 x C 4 x _ 4 x I 4 x 5$
%(0$
$$$$$$$$$$$$$$$$$$$$$$$$$$$ 0CW 4 x7 ! 08 4 x 7 ! 8 # 1 8 4 x 7 ! # x O # x C 4 x _ 4 x I 4 x 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$"$
$
$
:;.'(<&#!H#
Q"G#5$1"$#%K"$ n ! H _ $%(0$ q ! I 6$Q+1$1"$;%."$ r @! +/0 H _ 4I7 R_$%(0$ % ! * 48J7 S _ ! H 6$
D;"$:3:>+#+,':$:+*"#*$%/"$ C P ! APB 5$ C8 ! A85I5U588B 5$ C H ! AH5JB 5$ C _ ! A_58HB 5$
C C ! AC58C58I5OB 5$ CT ! ATB 5$ C8P ! A8P58_B 5$%(0$#;"$:+//"*-+(0'(&$'//"0?:')>"$-+>3(+,'%>*$
$
_I
%/"$ PP 4 x7 ! x # 8 5$ P8 4 x7 ! x _ # x H # 8 5$ PH 4 x7 ! x H # x # 8 5$ P_ 4 x7 ! x H 4 8 5$
PC 4 x7 ! x _ 4 x H # 8 5$ PT 4 x7 ! x 4 8 $%(0$ P8P 4 x 7 ! x H 4 x # 8 6$
V+/$#;"$*%,"$/"%*+($%*$'($#;"$-/".'+?*$"G%,->"5$#;"$(?,)"/$+2$'//"0?:')>"$2%:#+/*$'($
' T 4 x 7 $'*$?("F?%>$#+$#;"$(?,)"/$+2$'//"0?:')>"$2%:#+/*$'($ ' _ 4 x7 6$
D;"$2%:#+/'[%#'+($+2$ ' 8J 4 x7 $'(#+$#1+$'//"0?:')>"$-+>3(+,'%>*$+2$0"&/""$ r ! _ $'*$
' 8J 4 x 7 ! 4 x _ # x H # 874 x _ 4 x H # 87 6$D;"*"$-+>3(+,'%>*$%/"$#;"$0"2'('(&$-+>3(+,'%>*$+2$
C
8
#;"$"F?'.%>"(#$GRc:+0"*$ C8J
5 I5 H $%(0$ C8J5 I5 H 6$9"$(+1$;%."$*?22':'"(#$0%#%$#+$:+,-?#"$#;"$
">","(#*$+2$#;"$,%#/'G$ M 6$k*'(&$%&%'($4HO75$1"$2'(0$#;"$2+>>+1'(&$">","(#*$'($:+>?,($8@$
. P58 ! 8 5$ . H58 ! #8 5$ . T58 ! #8 5$ .8P 58 ! 8 5$ .858 ! . _58 ! . C58 ! P 6$<+5$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$$$$$$$$$$$$ 1 8 4 x 7 ! 8J #8 4c P 4 x 7 # c H 4 x 7 # cT 4 x 7 4 c8P 4 x 77 R 8 # x H # x J # x T 4 x 8P 4 x 8_ 5$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 08 4 x7 ! # x 8_ # x 8P 4 x T 4 x 8P 4 x 8_ 6$
$
<','>%/>35$1"$2'(0$
$
$$$$$$$$$$$$$ 1 C 4 x7 ! 8J #8 4c P 4 x7 4 c H 4 x7 # cT 4 x 7 # c8P 4 x 77 ! 8 4 x H 4 x J # x T # x 8P # x 8_ 5$
$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$ 0C 4 x 7 ! x 8_ 4 x 8P 4 x T # x J # x H 6$
$
=(0""05$#;"$'0",-+#"(#$&"("/%#+/*$+2$ C Hi _ 5I5 H $:%($)"$+)#%'("0$2/+,$#;+*"$+2$ C Hi I 5I5 H 5$
i $ A85CB 5$)3$/"->%:'(& x $)3$ x H 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$"$
$
$
G"##23&#1./&# n ! H p < #
D;/+?&;+?#$#;"$0'*:?**'+($+2$#;'*$:%*"$1"$%**?,"$%&%'($#;%#$:+(0'#'+($4W7$+2$-6$IU$;+>0*6$
9"$;%."$#;"$/">%#'+(*$ r @! $+/0 H p< 4q 7 ! +/0 p< 4q 7 5$ u @! +/0 p 4q 7 5$%(0$
% ! * 4 p 7 S u ! p # 8S u 6$=($%$*','>%/$1%3$%*$'($<":#'+(*$J$%(0$O5$1"$0"/'."$#;%#$#;"$
<
2%:#+/'[%#'+($+2$ x n # 8 R$ x H p # 8 $$'(#+$:3:>+#+,':$-+>3(+,'%>*$(+1$;%*$#;"$2+/,$
$
<
$$$$$ x H p # 8 ! '8 4 x7' H 4 x7' p 4 x7' p H 4 x7666666' p< 4 x7' H p 4 x7' H p H 4 x7666666' H p< 4 x7 $
$$$$$$$$$$$$$$$$$$R 4 x # 874 x 4 87' p 4 x7' p H 4 x766666' p< 4 x7' p 4 # x7' p H 4 # x766666' p< 4 #x7 5$$$$$$$$$$$$$$4JP7$
1'#;$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ' pb 4 x7 !
(
b#8
Pi 4 x p 7 5$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$4J8$7$
i$S fhAPB
$
2+/$%>>$b, 8 " b " < 5$%(0$1;"/"$#;"$ % $,+(':$-+>3(+,'%>*$ Pi 4x 7 5$ i $ S fh APB 5$%/"$#;"$
'//"0?:')>"$2%:#+/*$'($ GF 4 q 7\ x] $:+(#%'("0$'($ ' p 4x 7 6$`"/"5$$>'K"$'($<":#'+($O5$ S f $*#%(0*$
$
__
2+/$#;"$*"#$+2$'(0':"*$+2$#;"$:3:>+#+,':$:+*"#*$,+0$ p 6$D;"$,+(':$'//"0?:')>"$-+>3(+,'%>*$
< #8
:+(#%'("0$'($ ' H p< 4 x7 5$1;':;$:%($)"$1/'##"($%*$ 4#87 r Pi 4# x p 7 5$ i $ S fh APB 5$%/"$#;"$
,'(',%>$&"("/%#+/*$+2$#;"$GRc:+0"*$ CHi p< 5 q 5t $1'#;$ t ! % 6$D;"$:+>?,(*$+2$#;"$,%#/'G$M$
1;':;$:+//"*-+(0$#+$#;"*"$-+>3(+,'%>*$0"#"/,'("$#;"$'0",-+#"(#$&"("/%#+/*$+2$#;"*"$GRc
:+0"*6$
l?*#$>'K"$'($<":#'+($J5$1"$(+1$-/"*"(#$%$>",,%$1;':;$1'>>$"(%)>"$?*$#+$"*#%)>'*;$#;"$
/">%#'+(*;'-$)"#1""($#;"$%)+."$'//"0?:')>"$-+>3(+,'%>*$%(0$#;"$:3:>+#+,':$:+*"#*$,+0$
H p < $1'#;$/"*-":#$#+$q6$=($$!",,%$CP$$#;"$(+#%#'+($ S $*#%(0*$%&%'($2+/$#;"$*"#$+2$'(0':"*$
+2$:3:>+#+,':$:+*"#*$,+0$ p < 5$1;'>"$ S ff $0"(+#"*$#;"$*"#$+2$*?:;$'(0':"*$,+0$ H p < 6$
<','>%/>3$%*$'($<":#'+($J5$$1"$0"2'("$#;%#$'2$ Ci ! Ai5 iq5666665 iq rP #8B 5$ i $ S 5$ rP ! r 5$$'*$%$
:3:>+#+,':$:+*"#$,+0$ p < 5$#;"($ CHi $'*$#;"$*"#$:+(*'*#'(&$+2$%>>$'(#"&"/*$ Ha 5$ a $ Ci 6$D;'*$
*"#$ CHi $'*$%$:3:>+#+,':$:+*"#$,+0$ H p < 6$V?/#;"/,+/"5$1"$'(#/+0?:"$2+/$"%:;$ i $ S 5$%$*"#$
:+(*'*#'(&$+2$#;"$'(#"&"/*$ a $ Ci $1;"($a '*$+00$%(0$ a 4 p < $1;"($a '*$"."(6$=2$i '*$+005$1"$
:%>>$#;'*$*"#$%&%'( Ci 5$%(0$'2$ i $'*$"."($ Ci 4 p< 6$D;"$*"#*$ Ci $%(0$ Ci 4 p< 5$ i $ S 5$#+&"#;"/$
:+(#%'($%>>$+00$-+*'#'."$'(#"&"/*$1;':;$%/"$>"**$#;%($ H p < 6$$D;"$*"#*$ C Hi 5$ i $ S 5$#+&"#;"/$
:+(#%'($%>>$"."($(+(("&%#'."$'(#"&"/*$>"**$#;%($ H p < 6$$
$
$
E&''.#9I$
4'7$$ C Hi is a cyclotomic coset mod$$ H p < $for any$$ i $ S X
4''7$$ C i $and Ci 4 p< are$cyclotomic cosets mod H p < for all i $ S X$
4'''7$$if a $ C i , then mod H p < , Ha $ C Hi X$
4'.7$$$the cyclotomic$cosets in 4'7$and$4''7$together constitute the complete family of
H% 4 H cyclotomic cosets mod H p < and their indices constitute the index set S ff X$
4.7$$$the cyclotomic cosets in$4'7$correspond to irreducible polynomials contained in
' p< 4 x7 and those in 4''7$to irreducible polynomials contained in$ ' H p< 4 x7 X$
4.'7$$if Pi 4x7 is an irreducible factor of ' p< 4 x7 , then Pi 4 p< 4 x 7 is an irreducible factor of
' H p< 4 x7 and vice versa, for all i $ S and Pi 4 p< 4 x7 ! 4#87 r Pi 4# x7 X$
4.''7$$if C j 4 p < , j $ S and j odd, is the set of integers a $ C j for a even and a 4 p < for a
odd, then the family AC j 4 p < @ j $ S B is the same as the family AC H j @ j $ S B defined in 4i7X$
4.'''7$$ CHi ! Ci 4 p< , and hence PHi 4 x7 ! Pi 4 p< 4 x7 , for all i $ S , if and only if H $ C8 . $
D;"$-/++2*$%/"$*','>%/$#+$#;"$-/++2*$+2$!",,%$IO6$
$
<','>%/>3$#+$#;"$>%)">'(&$4CO75$1"$(+1$'(#/+0?:"$$#;"$2+>>+1'(&$&"("/%>$>%)">'(&$+2$#;"$
:+>?,(*$%(0$/+1*$+2$#;"$,%#/'G$M$$4:26$#;"$>%)">'(&$'($#;"$:%*"*$ n ! H p $%(0$ n ! p < 7@$
$
_C
$
$$$$$$$$$$$$$$$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$
$$$$$$$$$$$$$$$$$$ P5 i8< 5 iH< 5666666665 i%< 5 i8< #8 5 iH< #8 5666666665 i%< #8 5666666666665 i88 5 iH8 56666666665 i%8 5$
$$$$$$$$$$$$$$$$$$$$$ i%<48 5 i%<4 H 566665 iH<% 5 i%<4#88 5 i%<4#H8 566666665 iH<%#8 566666666665 i%8 48 5 i%8 4 H 566665 iH8% 5 p < $$$$$$$$$$$$$$$$$$$$$$$4JH7$$
$
D+&"#;"/5$#;"$'(0':"*$'($4JH7$:+(*#'#?#"$#;"$'(0"G$*"#$ S ff $+2$#;"$:3:>+#+,':$:+*"#*$,+0$
H p < 6$D;"$,?#?%>$0"-"(0"(:"$+2$#;"$'(0':"*$'($#;"$2'/*#$>'("$+2$4JH7$'*$&'."($)3$#;"$
/">%#'+(*$4CT7$%(0$4CU76$9'#;$/"*-":#$#+$#;"*"$/">%#'+(*5$+("$*;+?>0$$/",",)"/$#;%#$#;"$
'(0"G$+2$%$:3:>+#+,':$:+*"#$'*$(+#$?('F?">3$0"#"/,'("05$)?#$:%($'($-/'(:'->"$)"$"F?%>$#+$
%(3$+2$#;"$'(#"&"/*$:+(#%'("0$'($#;%#$-%/#':?>%/$:+*"#6$$
9"$:;++*"$#;"$>%)">*$'($4JH7$*?:;$#;%#$ i8b 5 iHb 566665 i%b $'(0':%#"$'($*+,"$+/0"/$#;"$ % $,+(':$
b#8
'//"0?:')>"$-+>3(+,'%>*$$ 4#87 rb Pi 4# x p 7 5 i $ S fhAPB 5$$:+(#%'("0$'($ ' H pb 4 x7 ! ' pb 4# x7 5$
b #8
1;'>"$ i%b 48 5 i%b4 H 566665 iHb% $'(0':%#"$#;"$,+(':$'//"0?:')>"$-+>3(+,'%>*$ Pi 4 x p 7 5$ i $ S fhAPB 5$
:+(#%'("0$'($ ' pb 4 x7 5$2+/$ 8 " b " < 6$=($+#;"/$1+/0*5$1"$:;++*"$#;"$'(0':"*$*?:;$#;%#$
b#8
Pib 4 x7 ! 4#87rb P% 4ib 4# x7 5$2+/$ 8 " j " % 5$%(0$;"(:"$ Pib 4 x7 ! Pi j 4 x p 7 ! 4 #87 rb P% 4ib 4 # x7 6$
j
j
j
j
<
E::+/0'(&$#+$!",,%$CP$4.'75$#;'*$:%($)"$%::+,->'*;"0$)3$#%K'(&$ i% 4 j ! i j 4 p $5$2+/$
8 " j " % 5$*'(:"$ )
p<
! #8 $'2$ ) $'*$%$-/','#'."$ H p <
2+/$ ) $%$["/+$+2$ Pi< 4 x7 4R$ P8 4 x
8
p
< #8
th
$/++#$+2$?('#36$9"$*;%>>$%>1%3*$#%K"$
<
7 7$%*$-/','#'."$ H p $/++#$+2$?('#35$%(0$*+$ i8< ! 8 6$D;"$
2'(%>$'(0"G$ p < $'($4JH7$'(0':%#"*$#;"$:+>?,($:+//"*-+(0'(&$#+$#;"$:3:>+#+,':$:+*"#$
C p< ! A p < B 6$$
D;"$:+>?,(*$+2$ M $1'#;$%$>%)">$ i $ A i8< 5 iH< 5666665 i%< B$0"#"/,'("$#;"$'0",-+#"(#$&"("/%#+/*$
+2$$#;"$GRc:+0"*$ CHi p< 5 q 5t 6$E*$2+/$#;"$,%#/'G$">","(#*$'($#;"*"$:+>?,(*5$1"$;%."$*','>%/>3$
%*$'($#;"$:%*"$+2$ n ! p < 5$#;%#$ . j 5i ! P $?(>"**$ j $ Ai88 5666665 i%8 5 i%8 48 56666665 iH8% B +/$ j $ AP5 p < B 6$=($
-%/#':?>%/5$ . P 5i ! r $2+/$#;"*"$ i # .%>?"*$%(0$ . p< 5i ! #r 6$=#$1'>>$)"$+).'+?*$#;%#$#;"$'(0':"*$
4_I7$2+/,$%$*-":'%>$:%*"$+2$4JH7$$2+/$ < ! 8 6$9"$*;%>>$&'."$%($"G%,->"$#+$'>>?*#/%#"$#;"$
%)+."$:+(*'0"/%#'+(*6$
$
:;.'(<&#9J$
9"$#%K"$ n ! 8T RH6 I H $%(0$ q ! C 6$D;"($1"$;%."$ p ! I 5$ < ! H 5$ u @! +/0 I 4C7 ! H 5$
+/0 U 4C7 ! J $%(0$ r @! +/0 8T 4C7 ! J 6$=#$2+>>+1*$#;%#$ % ! * 48T7 S r ! 8 5$*+$#;"/"$'*$+(>3$+("$
8
GRc:+0"$ C8T
5 C58 6$$
D+$'>>?*#/%#"$#;"$/">%#'+(*;'-$)"#1""($#;"$:3:>+#+,':$:+*"#*$,+0$8T$%(0$,+0$U5$1"$2'/*#$
0"#"/,'("$#;"$>%##"/$:+*"#*@$ CP ! APB 5$ C8 ! A85 H5 _5C5 O5TB $%(0$ CI ! AI5 JB 6$E-->3'(&$
!",,%$CP$4'75$4''7$3'">0*$#;"$:+*"#*$,+0$8T@$ C P ! APB 5$
C8 ! A85C5O58858I58OB 5 CH 4! C8P 7 ! AH5 _5T58P58_58JB 5$ C I ! AI58CB 5$ CJ 4! C8H 7 ! AJ58HB $%(0$
$
_J
C U ! AUB 6$D;"$>%)">*$ i <j $+2$#;"$/+1*$%(0$:+>?,(*$+2$ M %/"$ <8H ! 8 5$ <HH ! I 5$ <H8 ! H 4+/$8P7$
%(0$ <H8 ! J 4+/$8H76$
D;"$2%:#+/'[%#'+($+2$ x 8T # 8 $'(#+$:3:>+#+,':$-+>3(+,'%>*$'*$
$
$$$$$$$$$$$$$$$$$$$$$$$$$ x 8T # 8 ! ' 8 4 x 7' H 4 x 7' I 4 x 7' J 4 x 7' U 4 x 7' 8T 4 x 7 $
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$R$ 4 x # 874 x 4 874 x H 4 x 4 874 x H # x 4 874 x J 4 x I 4 874 x J # x I 4 87 6$
$
E>>$#;"*"$:3:>+#+,':$-+>3(+,'%>*$%/"$'//"0?:')>"$'($ GF 4C7\ x ] $%(0$;"(:"$ PP 4 x7 ! x # 8 5$
P8 4 x7 ! x J # x I 4 8 5 PH 4 x7 ! x J 4 x I 4 8 5 PI 4 x 7 ! x H # x 4 8 PJ 4 x7 ! x H 4 x 4 8 5
PU 4 x7 ! x 4 8 5$%*$+("$:%($"%*'>3$."/'236$
$V+/$#;"$,%#/'G$ M $1"$2'(0$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$P$$$$8$$$$I$$$H$$$$J$$$$U$
58 8 H 8 H 8 6 P
7
8
78 P 8 P _ _ 8 8
78 I I H H _ 8 I
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ M ! 7
8 6$
78 P _ P _ 8 8 H
78 H H H H 8 8 J
77
88
98 _ I 8 H _ : U
$
8
D;"$'0",-+#"(#$&"("/%#+/$+2$ C8T
5 C58 $'*$&'."($)3$
$
$$$$$$$$$$$$$$$$$$$ 08W 4 x 7 ! 08 4 x 7 ! 8 # 18 4 x 7 ! 8 # 8T#8 4cP 4 x 7 4 IcI 4 x 7 4 HcJ 4 x 7 4 _cU 4 x77 $
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$R$ 8 # H48 # H4 x I 4 x8C 7 4 H4 x J 4 x8H 7 # xU 7 $
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$K# # x8C 4 x8H 4 H x U 4 x J # x I # 8 "##################################################!#
#
#
:;.'(<&#95$
Q"G#5$1"$#%K"$ n ! 8T $%(0$ q ! O 6$Q+15$#;"$-+>3(+,'%>*$ ' I 4 x7 5$ ' J 4 x7 5$ ' U 4 x7 $%(0$
'8T 4 x7 $%/"$(+$>+(&"/$'//"0?:')>"5$%(0$1"$:%($1/'#"$$
$
$$$$$$$$$$$ x8T # 8 ! '8 4 x7' H 4 x 7' I 4 x 7' J 4 x 7' U 4 x 7'8T 4 x 7 $
######################K 4 x # 874 x 4 874 x # _74 x # H74 x 4 _74 x 4 H74 xI # _74 xI # H74 xI 4 _74 xI 4 H7 ?#
#
1;"/"$#;"$-+>3(+,'%>*$'($#;"$/;*$%/"$%>>$'//"0?:')>"$'($ GF 4O7\ x ] 6$
D;"$:3:>+#+,':$:+*"#*$,+0$U$%/"$ CP ! APB 5$ C8 ! A85 _5 OB 5$ CH ! AH5C5TB 5$ CI ! AIB $%(0$
CJ ! AJB 6$$D;"$:+//"*-+(0'(&$'//"0?:')>"$-+>3(+,'%>*$%/"$ PPf 4 x7 ! x # 8 5$ P8f 4 x 7 ! x I # _ 5$
PHf 4 x 7 ! x I # H 5$ PIf 4 x 7 ! x # _ $%(0$ PJf 4 x 7 ! x # H 6$
$
_O
$
E-->3'(&$!",,%$CP$4'7$%(0$4''75$1"$+)#%'($#;"$:3:>+#+,':$:+*"#*$,+0$8T@$
CP ! APB 5$ C8 ! A85 O58IB 5$ CH ! AH5T58_B 5$ CC ! AC58858OB 5$ C_ ! A_58P58JB 5$ CI ! AIB 5$
CJ ! AJB 5$ C8H ! A8HB 5$ C8C ! A8CB $%(0$ CU ! AUB 5$
E*$-/','#'."$8T th $/++#$+2$?('#3$'($*+,"$"G#"(*'+($2'">0$+2$ GF 4O7 $+2$0"&/""$I5$1"$#%K"$2+/$
) $%$["/+$+2$ x I 4 _ ! x I # I 6$$Q"G#5$)3$%-->3'(&$ ) I ! I 5$1"$2'(0$$$
PP 4 x7 ! x # 8 5$ P8 4 x 7 ! x I # I 5 PH 4 x 7 ! x I # H 5$ PC 4 x 7 ! x I 4 H 5$ P_ 4 x 7 ! x I # _ 5$ PI 4 x7 ! x # I 5$
PJ 4 x7 ! x # H 5$ P8H 4 x7 ! x 4 I 5$ P8C 4 x7 ! x 4 H $%(0$ PU 4 x7 ! x 4 8 6$$
E*$+("$:%($."/'235$#;'*$*"#$+2$$'//"0?:')>"$-+>3(+,'%>*$:%($)"$+)#%'("0$2/+,$#;+*"$$2+/$
n ! U 5$)3$#%K'(&$ APi f 4 x 7 ! i $ S fhAPBB $ ; $Am$ Pi f 4 # x 7 ! i $ S fhAPBB 5$1;"/"$#;"$*'&(*$'($#;"$
*":+(0$*?)*"#$0"-"(0$+($#;"$0"&/""$+2$#;"$-+>3(+,'%>$ Pi f 4 # x 7 `"(:"5$#;"$'(0':"*$
P5 i8H 5 iHH 5 i88 5 iH8 5 iIH 5 i_H 5 iI8 5 i_8 5 p m $+2$4JH75$$;%."$#;"$.%>?"*$$ P585 C5 I58C5 _5 H58H5 J5 U 5$/"*-":#'.">36$
i
D;"$'0",-+#"(#$&"("/%#+/*$+2$#;"$GRc:+0"*$ C8T5O5H
5$$ i $ A85 CB 5$%/"$0"#"/,'("0$)3$#;"$
:+>?,(*$+2$#;"$,%#/'G$ M $1'#;$>%)">$8$%(0$C6$D;"$">","(#*$'($#;"*"$:+>?,(*$%/"$4:26$4HO77$
m
. j 5i ! # i pij 58 5$1'#;$ i $"F?%>$#+$8$%(0$C5$/"*-":#'.">36$9"$2'(0$2+/$#;"$:+//"*-+(0'(&$
mij
:+>?,($.":#+/*$ .8 ! 4I5 P5 P5 H585 P5 P5 #H5 #85 #I7T $%(0$ .C ! 4I5 P5 P585 H5 P5 P5 #85 #H5 #I7T 6$
8
C
`"(:"5$#;"$'0",-+#"(#$&"("/%#+/*$+2$ C8T5O5H
$$%(0$ C8T5O5H
$%/"$/"*-":#'.">3$
$
$$$$$$$$$$ 08W 4 x7 ! 8 # 1 C 4 x 7 ! 8 # 8T#8 4IcP 4 x 7 4 cI 4 x 7 4 Hc8C 4 x 7 # c8H 4 x 7 # HcJ 4 x7 # IcU 4 x 77 $
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ! 8 # H4I 4 x I 4 H x8C # x8H # H x J # I xU 7 $
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ! I x8C 4 H x8H # x U # I x J # H x I 4 H $
%(0$
$$$$$$$$$$$$ 0CW 4 x 7 ! 8 # 18 4 x 7 ! 8 # 8T#8 4IcP 4 x 7 4 HcI 4 x 7 4 c8C 4 x7 # Hc8H 4 x7 # cJ 4 x7 # IcU 4 x77 $
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ! 8 # H4I 4 H x I 4 x8C # H x8H # x J # I xU 7 $
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ #H x8C # I x8H # x U 4 H x J # _ x I 4 H 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$$
V+/$/"%*+(*$+2$:+,->"#"("**5$1"$2'(%>>3$-/"*"(#$#;"$:+,->"#"$,%#/'G$M @$
$
$
$
$
$
$
$
$
$
$
$
$
$
_T
$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$P$$$$$$8$$$$$$C$$$$$$I$$$$$8C$$$$$_$$$$$$H$$$$$8H$$$$J$$$$$$U$
$
8 8 I I
8 8 86 P
58 I I
7
8
I #H P P #I H #88 8
78 P P
7 8 P P #H
I P P H #I #8 8 C
7
8
8 #8 #8 #H #8 8 8 #88 I
78 H
78 8 H #8 #8 #8 #H
8 8 #88 8C
$$$$$$$$$$$$$$$$$ M ! 7
8 $$ $
78 P P #I H P P #I H 88 _
78 P P H #I P P H #I 88 H
7
8
78 #H #8 8 8 #H #8 8 8 88 8H
7
8 8 #8 #H
8 8 888 J
78 #8 #H
7
8 8 #8:8 U
98 #I #I #8 #8 I I
#
#
Q+#':"$#;%#$#;"$ Jth $:+>?,($+2$M :+//"*-+(0*$#+$#;"$:3:>+#+,':$:+*"#$ C84U ! C8P ! C_ 5$%(0$
#;"$ Oth $:+>?,($#+$ CC4U ! C8_ ! CH 6$
$$
Q"G#5$1"$-/"*"(#$%$#;"+/",$+($#;"$/">%#'+(*;'-$)"#1""($#;"$'0",-+#"(#$&"("/%#+/*$+2$GRc
:+0"*$+2$>"(&#;$ H p < $%(0$>"(&#;$ H p a $1'#;$ a " < $1;':;$'*$*','>%/$#+$D;"+/",$_C$6$9"$
*;%>>$(""0$#;"$2+>>+1'(&$>",,%6$
$
$
E&''.#94$
Let p be an odd prime and q some odd prime power with 4 p5 q 7 ! 8 . Let furthermore a be
some integer B 8 . The inequality ord H p a48 4q7 = ord H p a 4q7 implies
ord H p a4i#8 4 q 7 ! p i #8ord H pa 4 q 7 , for all i B 8 .
D;"$-/++2$'*$*','>%/$#+$#;"$2'/*#$-%/#$+2$$#;"$-/++2$+2$!",,%$$_8$$6$!'K"$!",,%$$_85$#;"$
%)+."$/"*?>#$'*$%>*+$%$*-":'%>$:%*"$+2$\H5$!",,%$$I6I_]$6$
$
=($#;"$("G#$#;"+/",5$#;"$*"#$ S a $'*$#;"$*"#$+2$#;"$'(0':"*$+2$#;"$:3:>+#+,':$:+*"#*$,+0$ H p a $
2+/$*+,"$ a B 8 5$1;'>"$ S ff $'*$#;"$*"#$+2$*?:;$'(0':"*$,+0$ H p < 6$<','>%/>3$%*$'($<":#'+($O5$
S a $'*$%$*?)*"#$+2$ S ff $%(0$ S a & U H p a $'*$%$*?)*"#$+2$ S ff& U H p< 5$'2$ a " < 6$
$
23&)-&'#9!$
Let a be some integer satisfying 8 " a " < . Let 0iW 4 x7 , i $ S ff& U H p< , be the idempotent
generator of the GR-code CHi p< 5 q 5t , and let 0iaW 4 x7 be the idempotent generator of CHi pa 5q 5t ,
$
_U
< #a
i $ S a & U H p a . If ord H p a48 4q 7 = ord H p a 4q7 , then 0iW 4 x 7 ! 0iaW 4 x p 7 .
$
D;"$-/++2$'*$:+,->"#">3$*','>%/$#+$#;"$-/++2$+2$D;"+/",$_C6$
9"$*;%>>$'>>?*#/%#"$D;"+/",$C_$)3$%($"G%,->"6$
$
:;.'(<&#99#
E&%'($1"$#%K"$ n ! 8T $%(0$ q ! O 6$=($#;"$-/".'+?*$"G%,->"$1"$:+,-?#"0$#;"$'0",-+#"(#*$
+2$#;"$GR-:+0"*$ C8Ti 5O5 H 5$ i $ A85CB 6$Q+15$1"$*;%>>$0+$#;"$*%,"$2+/$#;"$GR-:+0"*$ CJi 5O5 H 5$
i $ A85CB 6$V+/$ n ! J $%(0$ q ! O 5$1"$;%."$#;"$:3:>':$:+*"#*$ C P ! APB5 C8 ! A8B5 C H ! AHB5 $
C I ! AIB5 C _ ! A_B5 C C ! ACB6 $Q+#':"$#;%#$ S 8 ! AP585 H5I5 _5CB $'*$%$*?)*"#$+2$
S ff ! AP585 C5 I58C5 H5 _58H5 5 J5 UB $%(0$#;%#$ S 8 & U J ! S ff& U8T ! A85CB 6$$
9"$:%($1/'#"$$
$
$$$$$$$$$$$$$$$ x J # 8 ! 4 x # 87' H 4 x7' I 4 x7' J 4 x7 ! 4 x # 874 x 4 874 x H 4 x 4 874 x H # x 4 87 6$
$
<'(:"$ u @! +/0 J 4O7 ! 8 $%(0$ % ! H S u ! H 5$1"$:%($2%:#+/'["$ ' J 4 x7 $'($ GF 4O7\ x ] $%*$
x H # x 4 8 ! 4 x # I74 x 4 H7 6$<','>%/>35$1"$;%."$ x H 4 x 4 8 ! 4 x # H74 x 4 I7 6$V+/$%$-/','#'."$
J th /++#$+2$?('#3$1"$:;++*"$ ) ! I 6$`"(:"5$ P8f 4 x7 ! x # I 5$%(0$:+(*"F?"(#>3$ PHf 4 x7 ! x # H 5$
PIf 4 x 7 ! x 4 8 5$ P_f 4 x7 ! x 4 I 5$ PCf 4 x7 ! x 4 H $%(0$ PPf 4 x 7 ! x # 8 6$
k*'(&$#;"$:+(."(#'+($+2$>%)">'(&$4JH7$%(0$#;"$0"2'('#'+($+2$#;"$,%#/'G$">","(#*$ .i 5 j $'($
4HO75$1"$+)#%'($2+/$#;"$,%#/'G$ M f $'($#;'*$:%*"$
$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$P$$$$8$$$$$$C$$$$$$_$$$$$$H$$$$$I$
58 8 8 8 8 8 6 P
7
8
78 I #H #I H #88 8
78 #H I H #I #88 C
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ M f ! 7
8 6$
7 8 #I H #I H 8 8 _
78 H #I H #I 8 8 H
77
88
98 #8 #8 8 8 #8: I
$
`"(:"5$$
$
$$$$$ 188W 4 x 7 ! 1 Cf 4 x 7 ! J#8 48 # H x 4 I x C # x I # I x H 4 H x _ 7 R$ # I x C # H x _ 4 x I 4 I x H 4 H x # 8 5$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 088W 4 x7 ! I x C 4 H x _ # x I # I x H # H x 4 H 6$
$
<'(:"$+/0 8T 4O7 RI$e$+/0 J 4O7 ! 8 5$1"$%/"$%>>+1"0$#+$%-->3$D;"+/",$C_$1;':;$3'">0*$$
$
$
CP
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 08 4 x7 ! I x 8C 4 H x 8H # x U # I x J # H x I 4 H 5$
$
%(0$1;':;$'*$'0"(#':%>$#+$#;"$"G-/"**'+($2+/$ 08W 4 x 7 $'($bG%,->"$CH6$
<','>%/>35$1"$2'(0$$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 0C8W 4 x 7 ! #H x C # I x _ # x I 4 H x H # _ x 4 H 5$
$
%(0$%&%'($)3$D;"+/",$C_$
$
W
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 0C 4 x7 ! #H x8C # Ix8H # x U 4 H x J # _ x I 4 H 5$
$
#;"$*%,"$"G-/"**'+($%*$2+?(0$'($bG%,->"$CH6$
$
N+,)'('(&$$D;"+/",*$IU$%(0$C_5$1+?>0$3'">0$"G-/"**'+(*$2+/$#;"$'0",-+#"(#$&"("/%#+/*$
+2$:+0"*$+2$>"(&#;$ H p < $'($#"/,*$+2$'0",-+#"(#$&"("/%#+/*$+2$:+0"*$+2$>"(&#;$p 5$1;"($
a ! 8 6$9"$+,'#$*?:;$"G-/"**'+(*6$$
$
$
H"#L#0&M#-&'.-8/#)+#*3&#N+-&/*-O1*&%#1./&/#n = p < and n = 2p < $
$
=($#;'*$*":#'+($1"$)/'"2>3$0'*:?**$#;"$*'#?%#'+($1;"($:+(0'#'+($4W7$'*$(+#$*%#'*2'"06$
$
:;.'(<&#9=#
D%K"$ n ! CH $%(0$ q ! O 6$9"$(+1$;%."$+/0 C 4O7 ! $+/0 HC 4O7 ! _ $%(0$+/0 8HC 4O7 ! HP 6$$<+5$
r8 ! _ 5$ rH ! _ $%(0$ rI ! HP 6$`"(:"5$#;"$'(#"&"/$a '($!",,%$_8$:%((+#$)"$#%K"($"F?%>$#+$86$
W
=#*$,'(',%>$.%>?"$'*$H6$V?/#;"/,+/"5$ %8 ! * 4C7 S r8 ! 8 5$ % H ! * 4HC7 S rH ! C $%(0$
% I ! * 48HC7 S rI ! C 6$
$
D;"$:3:>+#+,':$:+*"#*$,+0$C$%/"$ CP ! APB $%(0$ C8 ! A85 H5 _5IB $1'#;$:+//"*-+(0'(&$
'//"0?:')>"$-+>3(+,'%>*$4'($ GF 4O7\ x ] 7$ PP 4 x7 ! x # 8 %(0$ P8 4 x 7 ! x _ 4 x I 4 x H 4 x 4 8 6$
8
D;"$*":+(0$-+>3(+,'%>$&"("/%#"*$#;"$GR-:+0"$ CC5O58
6$
$
$D;"$:3:>+#+,':$:+*"#*$,+0$HC$%/"$ CP ! APB 5$ C8 ! A85 O5 H_58TB 5$ CH ! AH58_5 HI588B 5$
CI ! AI5 H85 HH5 _B 5$ CJ ! AJ58O58U5TB 5$ CU ! AU58I58J58HB $%(0$ CC ! AC58P5 HP58CB 6$D;"$
'//"0?:')>"$-+>3(+,'%>*$:+//"*-+(0'(&$#+$#;"*"$:+*"#*$%/"$/"*-":#'.">3$ PP 4 x7 ! x # 8 5$
P8 4 x 7 ! x _ 4 H x I 4 _ x H 4 H x 4 8 5$ PH 4 x 7 ! x _ 4 _ x I 4 I x H 4 _ x 4 8 5$ PI 4 x 7 ! x _ 4 _ x I 4 _ x 4 8 5$
PJ 4 x 7 ! x _ 4 C x I 4 C x H 4 C x 4 8 5$ PU 4 x 7 ! x _ 4 J x I 4 C x H 4 J x 4 8 $%(0$
PC 4 x7 ! x _ 4 x I 4 x H 4 x 4 8 6$
V/+,$#;"*"$0%#%$1"$0"/'."$#;"$"G->':'#$2+/,$+2$#;"$,%#/'G$M5$1;':;$%--"%/*$#+$)"$
$
$
C8
$
$
$
$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$P$$$$$$8$$$$$$H$$$$$$I$$$$$$J$$$$$$U$$$$$C$
58 _ _ _ _ _ _ 6 P
7
8
7 8 #H #_ #_ #C #J #88 8
7 8 #_ #_ #C #J #H #88 H
7
8
M ! 7 8 #_ #C #J #H #_ #88 $ I 6$
7 8 #C #J # H # _ # _ # 8 8 J
7
8
7 8 #J #H #_ #_ #C #88 U
7 8 #8 #8 #8 #8 #8 _ 8 C
9
:
$
$
D;"$'//"0?:')>"$-+>3(+,'%>*$ P8 4 x7 5$ PH 4 x 7 5$ PI 4 x 7 5$ PJ 4 x7 $%(0$ PU 4 x7 $&"("/%#"$#;"$GRi
:+0"*$ CHC5O5C
5$ i $ A85 H5 I5 J5 UB 5$1;'>"$#;"$:+>?,(*$1'#;$#;"*"$'(0':"*$0"#"/,'("$#;"$
'0",-+#"(#$&"("/%#+/*$+2$#;"*"$:+0"*6$<+5$*','>%/$#+$#;"$(+#%#'+($4CO75$1"$(+1$1/'#"$
i8H ! 8 5$ iHH ! H 5$ iIH ! I 5$ i_H ! J 5$ iCH ! U $%(0$ i88 ! C 6$
#
9"$%>*+$'(."*#'&%#"$#;"$/">".%(#$0%#%$'($#;"$:%*"$ n ! CI 5$ q ! O 6$
D;"$:3:>+#+,':$:+*"#*$,+0$8HC$%/"$ CP ! APB 5$
C8 ! A85 O5 _U5UI5 HJ5CO5 H_5 _I5C858PO58H_588T5 OJ5IH5UU5 JT58885TH5 O_58TB 5$
CH ! HC8 5$ CI ! IC8 5$ CJ ! JC8 5$ CU ! UC8 5$ CC ! AC5IC58HP5UPB 5$ C8P ! A8P5 OP588C5CCB 5$
C8C ! A8C58PC588P5 HPB 5$ CIP ! AIP5TC5UC5 _PB 5 C_C ! A_C5 JC5TP5 JPB $%(0$
CHC ! AHC5CP58PP5 OCB 6$
<+5$'(*#"%0$+2$>%)">'(&$4CO75$1"$(+1$;%."$#;"$2+>>+1'(&$'(0':"*$2+/$#;"$:3:>+#+,':$:+*"#*$$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$P5$85$H5$I5$J5$U5$C5$8P5$8C5$IP5$_C5$HC6$
$
<','>%/>3$#+$#;"$(+#%#'+($+2$4CO75$1"$0"(+#"$$#;"*"$'(0':"*$)3$ i8I ! 8 5$ iHI ! H 5$ iII ! I 5$ i_I ! J 5$
iCI ! U 5$ i8H ! C 5$ iHH ! 8P 5$ iIH ! 8C 5$ i_H ! IP 5$ iCH ! _C 5$ i88 ! HC 6$D;"$'//"0?:')>"$-+>3(+,'%>*$
:+//"*-+(0'(&$#+$#;"$:3:>+#+,':$:+*"#*$%/"$/"*-":#'.">3$ PP 4 x74! x # 87 5$ P8 4 x C 7 5$ PH 4 x C 7 5$
PI 4 x C 7 5$ PJ 4 x C 7 5$ PU 4 x C 7 5$ PC 4 x 7 5$ P8P 4 x 7 5$ P8C 4 x 7 5$ PIP 4 x7 5$ P_C 4 x 7 $%(0$ PHC 4 x7 6$V/+,$ r @! $
+/0 CI 4O7 ! HP $%(0$ * 4CI 7 ! CH 4C # 87 ! 8PP 5$'#$2+>>+1*$#;%#$ % ! 8PP S HP ! C 6$D%K'(&$ t ! % 5$
&'."*$#;%#$#;"$C$:+>?,(*$+2$M >%)">"0$)3$85$H5$I5$J$%(0$U$0"#"/,'("$#;"$'0",-+#"(#$
&"("/%#+/*$+2$#;"$GR-:+0"*$ CCiI 5O5C 5$ i $ A85 H5 I5 J5 UB 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$" $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$
$
CH
D;"$-/".'+?*$"G%,->"$*?&&"*#*$#;"$2+>>+1'(&$"G#"(*'+($+2$!",,%$__$1;':;$+(>3$;+>0*$'2$
:+(0'#'+($4W7$'*$*%#'*2'"05$'6"6$'2$+/0 pH 4q 7 = $+/0 p 4q 7 $%(0$%00'#'+(%>>3$+/0 HI 4q 7 = $+/0 HH 4 q 7 $
2+/$ p ! H 6$
$
E&''.#9F$
Let p be an arbitrary prime and q some prime power such that 4 p5 q 7 ! 8 . Let furthermore
rb @! ord pb 4q7 and % b @! * 4 p b 7 S rb , for b B 8 . Then the following statements hold:
4'7$$$if p is odd, there is an integer d, d B 8 , with 8 " r8 ! 6666 ! rd > rd 48 > rd 4 H 6666 , $ rb 48 ! prb $$
for b B d ,$and$8 " %8 > 6666 > % d ! % d 48 ! % d 4 H ! 6666 5$$ % b 48 ! p% b $for 8 " b > d X$
4''7$$if p = 2, there is an integer d, d B H , such that either
8 ! r8 ! rH ! 6666 ! rd > rd 48 > rd 4 H > 6666 and 8 ! %8 > % H > 6666 > % d ! % d 48 ! % d 4 H ! 6666 or
8 ! r8 > H ! rH ! 6666 ! rd > rd 48 > rd 4 H > 6666 and 8 ! %8 ! % H > 6666 > % d ! % d 48 ! % d 4 H ! 6666 , while
rb 48 ! Hrb for b B d , and % b 48 ! H% b for 8 " b > d and H " b > d respectively X
4'''7$$there are % b nonzero cyclotomic cosets mod p b and of size rb for b B 8 X$
4'.7$$to each cyclotomic coset Ci ! Ai5 qi566665 q rb #8iB mod p b of size rb with 4i5 p 7 ! 8 , there
corresponds a cyclotomic coset Ci ! Ai5 qi566665 q rb48 #8iB mod pb 48 of size rb 48 , for b B 8 X$
4.7$$$to each cyclotomic coset Ci ! Ai5 qi566665 q rc #8iB mod p b of size rc , 8 " rc " rb , there
corresponds a cyclotomic coset C pi ! A pi5 pqi566665 pq rc #8iB mod p b 48 , also of size rc , for
b B 8 6$
$
6-))0$
^">%#'+(*$4'7$%(0$4''7$2+>>+1$',,"0'%#">3$2/+,$!",,%$_85$4'''7$'*$%$:+(*"F?"(:"$+2$#;"$
/">%#'+(*$ % b ! * 4 p b 7 S rb $%(0$ * 4 p b 7 ! p b #8 4 p # 87 5$4'.7$2+>>+1*$2/+,$ rb 48 ! rb $2+/$$ 8 " b > d 5$
%(0$2/+,$ rb 48 ! prb $2+/$ b B d $'($4'75$1;'>"$/">%#'+($4.7$'*$+).'+?*6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$"$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$
$
$
<','>%/$%*$"F6$4CH75$#;"$:3:>+#+,':$-+>3(+,'%>$ ' pb 4 x7 $'*$#;"$-/+0?:#$+2$ % b $'//"0?:')>"$
-+>3(+,'%>*$'($ GF 4 q 7\ x] 5$2+/$ b B 8 5$1;':;$%/"$%>>$+2$0"&/""$ rb 5$1;"/"$#;"$'(#"&"/*$ % b $
%(0$ rb $%/"$%*$0"*:/')"0$'($!",,%$CO6$V+/$ b B d 5$#;"$'//"0?:')>"$-+>3(+,'%>*$ Pib 4 x7 $:%($
j
)"$"G-/"**"0$'($#"/,*$+2$#;"$-+>3(+,'%>*$ Pid 4 x7 5$%::+/0'(&$#+$$
j
b# d
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ Pib 4 x 7 ! Pi d 4 x p 7 5$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
j
j
4JI7$
$
%*$%$:+(*"F?"(:"$+2$!",,%$CO$4.76$4N+,-%/"$#;"$/",%/K*$/'&;#$%2#"/$4CO7$%)+?#$#;"$
b #8
-+>3(+,'%>*$ Pi 4 x p 7 5$ i $ S fh APB 5$1;"/"$d$R$867$=2$ n ! p < 5$#;"$:+>?,(*$+2$#;"$,%#/'G$ M 5$
1;':;$:+//"*-+(0$#+$#;"*"$'//"0?:')>"$-+>3(+,'%>*5$:%($)"$>%)">"0$'($&"("/%>$)3$
$
$
CI
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ P5 i8< 5 iH< 566665 i%<< 5 i8< #8 5 iH< #8 566665 i%<<##88 5666666665 i88 5 iH8 56666i%88 5$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
4J_7$$
$
1'#;$ i8< ! 8 6$9"$/",%/K$#;%#$>%)">'(&$4J_7$'*$#;"$*%,"$%*$>%)">'(&$4CO7$'($:%*"$#;%#$
:+(0'#'+($4W7$;+>0*6$=($#;%#$:%*"$1"$;%."$'($!",,%$CO$#;%#$ d ! 8 $%(0$;"(:"$
%8 ! % H ! 66666 ! % < 4!@ % 7 6$=($#;"$&"("/%>$:%*"5$#;"$:+>?,(*$+2$ M $1'#;$>%)">*$ i8< 5 iH< 566665 i%<< $
0"#"/,'("$#;"$'0",-+#"(#$&"("/%#+/*$+2$#;"$ % < $GR-:+0"*$ C ip< 5q 5t 5$1'#;$ t ! % < $%(0$
i $Ai8< 5 iH< 566665 i%<< B 6$$$$
=($#;"$:%*"$#;%#$ n ! H p < 5$ < B 8 5$1"$;%."$#;%#$ ' H pb 4 x7 ! ' pb 4# x7 5$ 8 " b " < 5$$'*$%>*+$#;"$
-/+0?:#$+2$ % b $'//"0?:')>"$-+>3(+,'%>*$'($ GF 4 q 7\ x] 5$%>>$+2$0"&/""$ rb 5$1;':;$:%($)"$
+)#%'("0$2/+,$#;+*"$$'($ ' pb 4 x7 $)3$/"->%:'(&$ x $)3$ # x 6$=($#;'*$:%*"$#;"$>%)">'(&$+2$#;"$
,%#/'G$ M $:%($)"$%::+,->'*;"0$)3$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$ P5 i8< 5 iH< 566666666665 i%<< 5 i8< #8 5 iH< #8 56666666665 i%<<##88 566666665 i88 5 iH8 566666665 i%88 5 $
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ i%<< 48 5 i%<< 4 H 56665 iH<%< 5 i%<<##88 48 5 i%<<##88 4 H 56665 iH<%#<8#8 56666665 i%88 48 5 i%88 4 H 56665 iH8%8 5 p < 6$$$$$$$$$$$
4JC7$$
$
9"$:;++*"$#;"$'(0':"*$'($#;"$*":+(0$/+1$+2$4JC7$*?:;$#;%#$ Pib 4 x7 ! 4#87 rb P%
j
b
b 4i j
4# x7 5$2+/$
8 " b " < 5$4:26$%>*+$#;"$/",%/K*$%)+?#$#;"$'//"0?:')>"$-+>3(+,'%>*$ Pi 4 x7 $+($-6$_J76$E&%'($
#;"$>%)">*$ i8< 5$ iH< 5$g65 i%<8 $/"2"/$#+$&"("/%>'["0$/"*'0?"$:+0"*5$'6"6$#;"$:+0"*$ CHi p< 5 q 5t 5$1'#;$
t ! % < $%(0$ i $Ai8< 5 iH< 566665 i%<8 B 6$$
V+/$#;"$*-":'%>$:%*"$#;%#$ t ! % < ! H 5$1"$*#%#"$%(0$-/+."$#;"$2+>>+1'(&$-/+-"/#36$
$
E&''.#9G$
If % < ! H , then one has for the integer d in Lemma 57$@$
4'7$$$ d ! 8 for odd p$and < B 8 X$
4''7$ d ! H and < B H for p ! H and q ! 8 4 _l , l oddX$
$$$$$$$ d ! H $and < ! H for q ! 8 4 _l , l evenX$
4'''7$$ d ! I and < B I for p ! H and q ! #8 4 _l , l oddX$
$$$$$$$$ d ! I $and < ! I for p ! H and q ! #8 4 _l , l even.$$$
$
6-))0$
4'7$9"$*;%>>$*;+1$#;%#$2+/$+00$p %(0$ % < ! H $1"$%>1%3*$;%."$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ %8 ! % H ! 666666 ! % < ! H 6$
E**?,"$#;%#$#;'*$'*$(+#$#/?"6$V/+,$!",,%$CO$4'7$'#$#;"($2+>>+1*$#;%#$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 8 ! %8 > 6666 > % d ! % d 48 ! 6666 ! % < ! H 6$
$
C_
`"(:"5$ d ! H $%(0$ % H ! H 6$E*$%$:+(*"F?"(:"$1"$;%."$ r8 ! * 4 p 7 S8 ! p # 8 $%(0$
rH ! * 4 p H 7 S H ! p 4 p # 87 S H 6$V/+,$#;"$2'/*#$"F?%>'#3$'#$2+>>+1*$#;%#$ q p #8 ! 8 4 kp $%(0$*+$
*'(:"$ p = H 5$ q p #8S H ! #8 4 lp 5$2+/$:"/#%'($'(#"&"/*$k %(0$l6$Z?#$#;"($1"$1+?>0$;%."$
p
5 p6
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ q p 4 p #87 S H ! 4#8 4 lp 7 p ! #, 7 84#lp 7i ! #8 $$,+0$ p H 6$
i !P 9 i :
D;'*$:+(#/%0':#*$#;"$/">%#'+($ rH ! p 4 p # 87 S H 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
4''7$$V+/$ p ! H $%(0$ q ! 8 4 _l 5$l +005$+("$:%($"%*'>3$*;+15$)3$:%>:?>%#'(&$#;"$.%>?"*$+2$ ri 5$
i B 8 5$%(0$%-->3'(&$ ri% i ! Hi #8 5$#;%#$
$$$$$$$$$$$$$$$$$$$$$$ 8 ! r8 ! rH > H ! rI > r_ > 6666 $$%(0$$ 8 ! %8 > H ! % H ! % I ! % _ ! 6666 $
=2$ q ! 8 4 _l 5$l "."(5$+("$+)#%'(*$*','>%/>3$
$$$$$$$$$$$$$$$$$$$$$$ 8 ! r8 ! rH ! rI " r_ " 6666 $%(0$$ 8 ! %8 > H ! % H > _ ! % I " % _ " 6666 $
D;"$/">%#'+(*$2+/$d $%(0$ < $(+1$2+>>+1$',,"0'%#">36$
4'''7$=($%$*','>%/$1%3$+("$+)#%'(*$2+/$ p ! H $%(0$ q ! #8 4 _l 5$l +005$#;%#$
$$$$$$$$$$$$$$$$$$$$$$ 8 ! r8 > H ! rH ! rI > _ ! r_ > 6666 $$%(0$$ 8 ! %8 ! % H > H ! % I ! % _ ! 6666 $$5$
9;'>"$2+/$ p ! H $%(0$ q ! #8 4 _l 5$l "."(5$+("$2'(0*$
$$$$$$$$$$$$$$$$$$$$ 8 ! r8 > H ! rH ! rI ! r_ " rC " 6666 $$%(0$ 8 ! %8 ! % H > H ! % I > % _ " % C " 6666 $$
E&%'(5$#;"$/">%#'+(*$2+/$d %(0$ < $2+>>+1$',,"0'%#">36$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$"$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$
$
$
:;.'(<&#9H$
9"$#%K"$ p ! H $%(0$ < ! H 5$*+$ n ! H H 5$%(0$ q ! I 6$<+5$1"$%/"$'($#;"$2'/*#$:%*"$+2$!",,%$CT$
4'''76$D;"$(+(["/+$:3:>+#+,':$:+*"#*$,+0$H H $%/"$ C8 ! A85IB 5$ CH ! AHB 6$$D;"'/$*'["*$
:+//"*-+(0$#+$#;"$+/0"/*$ rH ! +/0 HH 4I7 ! H $%(0$ r8 ! $+/0 H8 4I7 ! 8 5$/"*-":#'.">36$
V?/#;"/,+/"5$1"$;%."$ %8 ! * 4H7 S8 ! 8 $%(0$ % H ! * 4H H 7 S H ! 8 6$
<+5$'($ GF 4I7\ x] 5$ ' H 4 x7 $2%:#+/'["*$'(#+$ %8 ! 8 $'//"0?:')>"$-+>3(+,'%>$+2$0"&/""$ r8 ! 8 $
%(0$ ' _ 4 x7 $'(#+$ % H ! 8 $'//"0?:')>"$-+>3(+,'%>$+2$0"&/""$ rH ! H 6$=(0""05$1"$:%($1/'#"$
H
x H # 8 ! '8 4 x7' H 4 x7' _ 4 x7 ! 4 x # 874 x 4 874 x H 4 87 6$D;"$>%)">'(&$4J_7$)":+,"*$'($#;'*$:%*"$
P5 i8H 5 i88 ! P585 H 5$%(0$#;"$:+//"*-+(0'(&$-+>3(+,'%>*$%/"$ P8 4 x 7 ! x H 4 8 $%(0$ PH 4 x7 ! x 4 8 6$
$
Q"G#5$1"$$#%K"$ < ! I 6$D;"$(+(["/+$:3:>+#+,':$:+*"#*$,+0$ HI $%/"$ C8 ! A85IB 5$ CC ! AC5 OB 5$
CH ! AH5 JB $%(0$ C_ ! A_B 6$D;"$*'["*$:+//"*-+(0$#+$#;"$+/0"/*$ rI ! $+/0 HI 4I7 ! H 5$ rH ! $
+/0 HH 4I7 ! H $%(0$ r8 ! $+/0 H8 4I7 ! 8 6$9"$:%($1/'#" xT # 8 ! '8 4 x 7' H 4 x 7' _ 4 x 7' T 4 x 7 $
R 4 x # 874 x 4 874 x H 4 874 x _ 4 87 ! 4 x # 874 x 4 874 x H 4 874 x H 4 x # 874 x H # x # 87 6$V+/$#;"$>%)">'(&$
4J_7$1"$(+1$;%."$ P5 i8I 5 iHI 5 i8H 5 i88 ! P585C5 H5 _ $1'#;$:+//"*-+(0'(&$'//"0?:')>"$-+>3(+,'%>*$
P8 4 x7 ! x H 4 x # 8 5$ PC 4 x 7 ! x H # x # 8 5$ PH 4 x 7 ! x H 4 8 5$ P_ 4 x7 ! x 4 8 6$
$
$
CC
D;"$("G#$:%*"$'*$ < ! _ 6$D;"$(+(["/+$:3:>+#+,':$:+*"#*$,+0$ H _ $%/"$$ C8 ! A85I5U588B 5$
CC ! AC58C58I5 OB 5$ CH ! AH5 JB 5$ C_ ! A_58HB 5$ CT ! ATB 5$ C8P ! A8P58HB 6$D;"$*'["$+2$ C8 $%(0$
CC $'*$"F?%>$#+$ r_ ! $+/0 H_ 4I7 ! _ 6$V?/#;"/,+/"5 '8J 4 x7 ! xT # 8 ! 4 x _ 4 x H # 874 x _ # x H # 87 5$
1;"/"$#;"$#1+$-+>3(+,'%>*$'($#;"$/;*$%/"$'//"0?:')>"6$$`"(:"5$ P5 i8_ 5 iH_ 5 i8I 5 iHI 5 i8H 5 i88 $R$$
P585C5H58P5_5T$%(0$#;"$'//"0?:')>"$-+>3(+,'%>*$%/" P8 4 x 7 ! x _ 4 x H # 8 5$ PC 4 x 7 ! x _ # x H # 8 5$
PH 4 x 7 ! x H 4 x # 8 5$ P8P 4 x 7 ! x H # x # 8 5$ P_ 4 x 7 ! x H 4 8 $%(0$ PT 4 x7 ! x 4 8 6$$$
_#I
Q+#':"$#;%#$ P8 4 x7 ! Pi_ 4 x7 ! PiI 4 x H 7 ! PH 4 x H 7 $%(0$*','>%/>3$ PC 4 x 7 ! P8P 4 x H 7 5$%::+/0'(&$#+$
8
8
"F6$4JI7$1'#;$ d ! I $4:26$!",,%$CT$4'''776$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$"$
$
$
$
$
:;.'(<&#=I$$
D%K"$ p ! H 5$ < ! _ $%(0$ q ! C 6$<+5$1"$%/"$(+1$'($#;"$*":+(0$:%*"$+2$!",,%$CT$4'''76$9"$
2'(0$ r8 ! 85 $ rH ! 85 $ rI ! H5 $ r_ ! _ 5$%(0$;"(:"$ %8 ! 85 $ % H ! H 5$ % I ! H5 $ % _ ! H 5$1;':;$*;+1*$
#;%#$'(0""0$ d ! H 6$D;"$:3:>+#+,':$:+*"#*$,+0$ H _ $%/"$ C8 ! A85C5U58IB 5$ CI ! AI58C5885 OB 5$
CH ! AH58PB 5$ C_ ! A_B 5$ CJ ! AJ58_B 5$ CT ! ATB 5$ C8H ! A8HB 6$=($ GF 4C7\ x ] $1"$;%."$#;"$
2+>>+1'(&$2%:#+/'[%#'+($'(#+$'//"0?:')>"$-+>3(+,'%>*$
$
$$$$$$$$$$$$$$$$$$$$$$$ x8J # 8 ! '8 4 x 7 ' H 4 x7' _ 4 x7' T 4 x7'8J 4 x7 $
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$R 4 x # 874 x 4 874 x 4 H74 x 4 I74 x H 4 H74 x H 4 I74 x _ 4 H74 x _ 4 I7 6$
$
D;"$>%)">'(&$4J_7$)":+,"*$ P5 i8_ 5 iH_ 5 i8I 5 iHI 5 i8H 5 iHH 5 i88 $R$P585I5H5J5_58H5T$6$D;"$:+//"*-+(0'(&$
-+>3(+,'%>*$%/"$ PP 4 x7 ! x # 8 5$ P8 4 x 7 ! x _ 4 H 5$ PI 4 x 7 ! x _ 4 I 5$ PH 4 x7 ! x H 4 H 5$
PJ 4 x 7 ! x H 4 I 5$ P_ 4 x7 ! x 4 H 5$ P8H 4 x7 ! x 4 I $%(0$ PT 4 x7 ! x 4 8 6$E*$+("$:%($*""$
_# H
',,"0'%#">35$1"$;%."$#;"$2+>>+1'(&$/">%#'+(*$ P8 4 x7 ! Pi_ 4 x7 ! PiH 4 x H 7 ! P_ 4 x _ 7 5$
8
8
H
PI 4 x7 ! Pi_ 4 x7 ! PiH 4 x H 7 ! P8H 4 x _ 7 5$%(0$*','>%/>3$ PH 4 x 7 ! P_ 4 x H 7 5$ PJ 4 x 7 ! P8H 4 x H 7 6$
H
H
D;"*"$/">%#'+(*$'>>?*#/%#"$"F6$4JI75$*'(:"$ q ! 8 4 _68 $%(0$ d ! H $4:26$!",,%$CT76$
$
Q"G#5$1"$$#%K"$ p ! H5 $ < ! C5 $ q ! O 6$V+/$#;"$ ri $%(0$ % i 5$ 8 " i " < 5$$1"$2'(0$ r8 ! 85 $ rH ! H5 $
rI ! H5 $ r_ ! H5 $ rC ! _ $%(0$ %8 ! 85 $ % H ! 85 $ % I ! H5 $ % _ ! _5 $ % C ! _ 6$E::+/0'(&$#+$!",,%$CO$
4''7$1"$(+1$;%."$ d ! _6 $$
D;"$(+(["/+$:3:>+#+,':$:+*"#*$,+0$ HC $%/"$ C8 ! A85 O58O5 HIB 5$ CH ! AH58_B 5$
CI ! AI5 H858U5CB 5$ C_ ! A_5 HTB 5$ CJ ! AJ58PB 5$ CT ! AT5 H_B 5$ CU ! AU5I85 HC58CB 5$$
C88 ! A8858I5 HO5 HUB 5$ C8H ! A8H5 HPB 5$ C8J ! A8JB 5$ C8T ! A8T5IPB 5$ CHH ! AHH5 HJB 6$
=($ GF 4O7\ x ] $#;"$2%:#+/'[%#'+($$+2$ x IH # 8 $'(#+$'//"0?:')>"$-+>3(+,'%>*$'*$%*$2+>>+1*$
$
$
CJ
$$$$$$$$$$ x IH # 8 ! '8 4 x 7' H 4 x7' _ 4 x7' T 4 x7'8J 4 x7' IH 4 x7 $
$$$$$$$$$$$$$$$$$$$$$R 4 x # 874 x 4 874 x H 4 874 x H # _ x 4 874 x H 4 _ x 4 87 4 x H 4 _ x # 874 x H # x # 87 $
$$$$$$$$$$$$$$$$$$$$$R 4 x H # _ x # 874 x H 4 x # 87 4 x _ 4 _ x H # 87 4 x _ # x H # 87 4 x _ # _ x H # 87 4 x _ 4 x H # 87 6$
$
D;"$:+//"*-+(0"(:"$)"#1""($:3:>+#+,':$:+*"#*$%(0$'//"0?:')>"$-+>3(+,'%>*$'*$%*$2+>>+1*6$
=2$1"$:;++*"$%*$-/','#'."$ IHth $/++#$+2$?('#3$%$["/+$+2$$ P8 4 x 7 @! x _ # x H # 8 5$#;"($
PU 4 x7 ! x _ 4 x H # 8 5$ PI 4 x 7 ! x _ # _ x H # 8 5$ P88 4 x 7 ! x _ 4 _ x H # 8 5$ PH 4 x 7 ! x H # x # 8 5$
P8T 4 x 7 ! x H 4 x # 8 5$ PJ 4 x 7 ! x H # _ x # 8 5$ PHH 4 x 7 ! x H 4 _ x # 8 5$ P_ 4 x7 ! x H 4 _ x 4 8 5$
P8H 4 x 7 ! x H # _ x 4 8 5 PT 4 x 7 ! x H 4 8 5$ P8J 4 x7 ! x 4 8 6$
<+5$1"$;%."$#;"$2+>>+1'(&$2%:#+/'[%#'+(*@$ ' IH 4 x 7 ! x8J 4 8 ! P8 4 x 7 PU 4 x 7 PI 4 x 7 P88 4 x7 5$
'8J 4 x 7 ! xT 4 8 ! PH 4 x7 P8T 4 x 7 PJ 4 x 7 PHH 4 x7 5 ' T 4 x 7 ! x _ 4 8 ! P_ 4 x 7 P8H 4 x 7 5
' _ 4 x 7 ! x H 4 8 ! PT 4 x 7 5$ ' H 4 x7 ! x 4 8 ! P8J 4 x7 6$
$
D;"$>%)">'(&$4J_7$)":+,"*$
$
$$$$$$$$$$$$$$$$$$$$$ P5 i8C 5 iHC 5 iIC 5 i_C 5 i8_ 5 iH_ 5 iI_ 5 i__ 5 i8I 5 iHI 5 i8H 5 i88 ! P585U5I5885 H58T5 J5 HH5 _58H5T58J $6$$$$$$$$$$$"$
$
$
7&'.-8#=J#
D;"$>%*#$:%*"$'($bG%,->"$JP$0",+(*#/%#"*$#;%#$2+/$ < ! C $#;"$.%>?"$+2$ % < $'*$(+#$"F?%>$#+$H6$
=($:%*"$1"$;%0$0"2'("0$ < ! I $4%(0$ q ! O 7$'($$#;"$-/".'+?*$"G%,->"5$1"$1+?>0$;%."$
% < ! % I ! H 5$%(0$*+$1"$:+?>0$%-->3$$#;"$2'/*#$-%/#$+2$!",,%$CT$4''75$*'(:"$ q ! 8 4 H6I 6$$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$
JI"#23&#GR-1)%&/#)0#<&+,*3#5J#)P&-#GFD!C#
#
=($#;'*$*":#'+($1"$*;%>>$:+(*'0"/$%($"G%,->"$1;"/"$#;"$:+0"$>"(&#;$'*$(+#$+2$#3-"$ p < $+/$
i
5$
H p < 6$=($\85<":#'+($T]$1"$0"/'."0$#;"$'0",-+#"(#$&"("/%#+/*$+2$#;"$#1+$$GR-:+0"*$ CH85_5H
)3$%$,"#;+0$/">%#"0$#+$#;"$+("*$,"(#'+("0$'($<":#'+($H6$$9"$(+1$*;%>>$:+,-?#"$#;"$*%,"$
i
&"("/%#+/*$)3$,"%(*$+2$#;"$,%#/'G$M6$^",",)"/$#;%#$#;"$:+0"*$ CH85_5H
$%/"$(+#$Hc/"*'0?"$
:+0"*5$#;+?&;$t$"F?%>*$H6$
$$$$$$$$$$$$
:;.'(<&#=5#$
9"$:+(*'0"/$#;"$:%*"$ n ! H8 5$ q ! _ $4:26$\85$<":#'+($T]76$
D;"$:3:>+#+,':$:+*"#*$%/"$$$
$
CP ! APB5 $ C8 ! A85 _58JB5 $ CH ! AH5T588B5 $ CI ! AI5 J58HB5 $ CC ! AC58O5 HPB5 $ CO ! AOB5 $
CU ! AU58C58TB 5$$ C8P ! A8P58I58UB5 $ C8_ ! A8_B 6$
$
CO
$
D;"$'//"0?:')>"$-+>3(+,'%>*$:+//"*-+(0'(&$#+$#;"*"$:+*"#*$%/"$/"*-":#'.">3$
$
PP 4 x7 ! x 4 85 $ P8 4 x 7 ! x I 4 D x 4 85 PH 4 x 7 ! x I 4 D H x 4 85 $ PI 4 x 7 ! x I 4 x H 4 8 5$
PC 4 x 7 ! x I 4 D x H 4 85 $ PO ! x 4 D 5$$ PU 4 x 7 ! x I 4 x 4 8 5$ P8P 4 x 7 ! x I 4 D H x H 4 8 5$$
P8_ 4 x7 ! x 4 D H 6$
$
<'(:"$#;"$*'["*$+2$%>>$:3:>+#+,':$:+*"#*$%/"$+005$#;"$'(#"&"/*$ n si $%/"$%>>$"F?%>$#+$8$'($
GF 4 _7 5$%(0$*+$#;"$,%#/'G$">","(#$ M i 5 s $'*$"F?%>$#+$ p is 58 $2+/$%>>$ i5 s $ S 6$9"$2'(0$$$#;"$
2+>>+1'(&$,%#/'G6$
$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$P$$$$$8$$$$$$$H$$$$$I$$$$$C$$$$$$O$$$$$U$$$$8P$$$$$8_$
$
8 8 8
8 8 8
8 6 P
58 8
8
7
H
P 8 D D P D DH8 8
78 P
78 P
P 8 DH DH P D D 8 H
8
7
8 P P
8 8 P
8 8 I
78 8
$$$$$$$$$$$$$$$$$$$$$$$$$ M ! 778 D D H P P D H 8 P D 88 $$ C 6$
78 D D H 8 D H D 8 D D H 8 O
8
7
P 8 8
8 P 8
8 8 U
78 P
78 D H D P P D 8 P D H 8 8P
8
77
H
D 8 D D H 8 D H D 8: 8_
98 D
$
V+/$#;"$:+0"$1'#;$,'(',%>$&"("/%#+/$ g8487 4 x7 ! P8 4 x7 PH 4 x7 5$1"$+)#%'($#;"$'0",-+#"(#$
&"("/%#+/$ e8487 4 x7 ! , + s c s 4 x7 5$1;"/"$#;"$.":#+/$ + $'*$0"#"/,'("0$)3$$
s$S
$$$$$$$$$$$$$$$$$$$$$$$$ + ! - P 4 M- C 4 M- 8P $R$ - P 4 .C 4 .8P $R$ 4858585 P5 P585 P5 P587T 6$
D;"$:+0"$&"("/%#"0$)3$ g 84 H 7 4 x 7 ! PC 4 x7 P8P 4 x 7 $;%*$#;"$'0",-+#"(#$&"("/%#+/$ e84 H 7 4 x 7 $1'#;$
:+"22':'"(#$.":#+/$$
$$$$$$$$$$$$$$$$$$$$$$$$$$ + ! - P 4 M- 8 4 M- H R - P 4 .8 4 .H $R$ 485 P5 P5 P58585 P58587T 6$
$
<','>%/>35$1"$2'(0$2+/$#;"$:+"22':'"(#$.":#+/*$+2$#;"$'0",-+#"(#$&"("/%#+/*$ eH487 4 x 7 5$
eH4 H 7 4 x 7 5$ eI487 4 x 7 $%(0$ eI4 H 7 4 x 7 5$/"*-":#'.">3$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ + ! - P 4 M - C 4 M -8 ! 485 D 5 D H 585 D 58585 D H 587T 5$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ + ! - P 4 M -8P 4 M - H ! 485 D H 5 D 585 D H 58585 D 587T 5$
$
$
CT
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ + ! - P 4 M - C 4 M - H ! 485 D 5 D H 585 D H 5 P585 D 5 P7T 5$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ + ! - P 4 M -8P 4 M -8 ! 485 D H 5 D 585 D 5 P585 D H 5 P7T 6$
$
D;"*"$*'G$:+"22':'"(#$.":#+/*$3'">0$#;"$*%,"$'0",-+#"(#*$%*$0"/'."0$)3$%$0'22"/"(#$,"#;+0$
?*"0$'($\85$<":#'+($T]6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$"$
$
$
7&'.-8#=4$
9"$*#'>>$'(."*#'&%#"$#;"$F?"*#'+($'($1;':;$1%3$#;"$,%#/'G$M$5$%*$'(#/+0?:"0$'($<":#'+($I5$
'*$/">%#"0$#+$#;"$:;%/%:#"/$#%)>"$+2$%$2'('#"$%)">'%($+/$(+(%)">'%($&/+?-6$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$ $
$
7&0&-&+1&/$
$
86$<6a6$Y+0?("K+.5$E6$Z+d'>+.$%(0$E6l6$.%($n%(#"(5$Generalized Residue Codes5$D^$
HP8PcPP85$D'NN5$D'>)?/&$k('."/*'#35$1116?.#6(>S#'::$
$
H6$^6!6$!'0>$%(0$`6$Q'"0"//"'#"/5$Introduction to Finite Fields and their Applications$4/".6$
"0675$N%,)/'0&"$k('."/*'#3$o/"**5$N%,)/'0&"$8UUO6$
$
I6$l6`6$.%($!'(#5$Coding Theory5$<-/'(&"/cp"/>%&5$Q"1$q+/K5$8UO86$
$
_6$l6`6$.%($!'(#$%(0$V6l6$a%:9'>>'%,*5$Generalized Quadratic Residue Codes5$=bbb$
D/%(*6$=(26$D;"+/3$5!$48UOT75$OIP$r$OIO6$
$
C6$V6l6$a%:9'>>'%,*$%(0$Q6l6E6$<>+%("5$The Theory of Error-Correcting Codes5$Q+/#;$
`+>>%(0$o?)>6$N+,-%(35$E,*#"/0%,5$8UOO6$
$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$
$
$$$$$$$$$$$$$$$$$$$
$
$
$$$$$$$$$$$$$$$$$$$$$$$$ $
$
$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$$$$$$$$$$$$$$$$$$$$$
#
$
#
$
CU
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
###############################
#
$
$
JP