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t
Ⅰ.Fore
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nt
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l
as
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he
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ut
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c
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Ⅰ.1TheECA mappi
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om t
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t
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l
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170.
Ⅰ.2Themappi
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e
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t
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he
r
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nt
hi
ss
ubc
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as
swi
t
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r
ul
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r
s94,122,126,184.
Ⅱ.Thi
,i
,name
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st
hel
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s
tc
l
as
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l
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s53ECAswhos
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2,3,4,5,6,8,10,11,12,13,14,18,19,23,24,29,32,34,35,36,38,40,42,43,44,46,50,51,
57,58,72,76,77,108,128,130,132,136,138,140,142,146,152,160,162,168,172,178,
200,204,232.Fr
om t
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nt
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nf
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ni
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t
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28,33,78,104,134,156,164.
Ⅳ.I
nt
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as
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nf
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ni
t
e
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ymanyDEBsandi
t
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gul
ar
l
anguage.The
r
ear
e13ECAsi
nt
hi
sc
l
as
swi
t
hr
ul
enumbe
r
s7,9,22,25,26,37,41,54,56,
62,73,74,110.
de fghi
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anber
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C,-.yE/I
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-E1E2I
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5.1 Thes
ubc
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The
r
ehavebe
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hepas
tmanydi
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c
us
s
i
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15,30,45,60,90,105,106,150,154,170i
nl
i
t
e
r
at
ur
e.A c
l
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orECA
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ul
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sus
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nMat
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mat
i
c
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t
sps
e
udor
andom
[16,17]
e
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r
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l
lonl
ys
how how t
opr
ovet
hatf30 i
st
r
ul
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ur
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c
t
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ve.
numbe
rge
ne
r
at
or
.
Thedi
s
c
us
s
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onaboutot
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rECAsi
nt
hi
ss
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as
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i
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ar
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r
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Pr
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on5.1.Thee
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ss
ur
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c
t
i
ve.
.Thel
Pr
oof
oc
alr
ul
eofECA ofr
ul
e30i
s
000,101,110,111→ 0;001,010,011,100→ 1.
(6)
I
tc
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t
t
e
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i
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f30(a- 1,a0,a1)= a- 1 o max(a0,a1),
(7)
,ando i
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hi
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l
oc
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ehast
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As
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umet
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0 1
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he
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s
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obee
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t
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=1byus
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t
ar
t
i
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om t
i
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-1
t
-1
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-1
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ax
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at
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- 1,
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x- 1
1
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e
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mi
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uc
c
e
s
s
i
ve
l
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et
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gume
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her
i
ght
hand
x- 1
s
i
deof(|),t
hef
i
r
s
tt
woar
eznownal
r
e
ady.The
ns
t
ar
t
i
ngf
r
om a1
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us
tobt
ai
ne
dand
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0
0
,andus
a a,t
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as
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t
t
e
r
soft
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e
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i
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f
t
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ur
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t
i
vepr
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r
t
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e
(7),wec
ande
t
e
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mi
nea- 1 ,...,a- x s
uc
c
e
s
s
i
ve
l
y.Thust
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e
}ui
r
e
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nve
r
s
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s
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ai
ne
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~xampl
e5.2.I
ts
uf
f
i
c
e
st
oe
xpl
ai
nt
heabovepr
oofbyas
i
mpl
ee
xampl
e.I
ti
sznownt
hat
e
e
00110i
sanfi
nve
r
s
ei
mageoft
het
i
mes
e
r
i
e
s110,andwewantt
oobt
ai
nf
ur
t
he
rt
hef-
i
nve
r
s
ei
mageoft
het
i
mes
e
r
i
e
s1101.Fr
om t
wos
ubf
i
gur
e
si
nt
hef
ol
l
owi
ngf
i
gur
ewes
e
e
e
,bas
t
hat
e
dont
hee
vol
ut
i
onpr
oc
e
dur
eoff(00110)=110,byt
azi
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t
t
e
ratt
her
i
ght
e
ndoft
hef
i
r
s
tl
i
net
obee
i
t
he
r0or1,andpr
oc
e
e
di
ngac
c
or
di
ngt
hedi
r
e
c
t
i
onmar
ze
dby
,al
ar
r
ows
lot
he
rl
e
t
t
e
r
sc
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t
e
r
mi
ne
duni
}ue
l
y.Thes
ymbol↓ i
sus
e
dt
ode
not
et
he
pos
i
t
i
onofs
i
t
e0.
↓
↓
1001101
1001100
& 11100 ’
& 11101 ’
001
001
1
1
258
.Mat
.B
Appl
h.J.Chi
ne
s
eUni
v.Se
r
.20,No.3
Vol
n- 1
Re
mar
k.I
tc
anbes
hownf
r
om t
heabovedi
s
c
us
s
i
ont
hatt
he
r
ear
ee
xac
t
l
y2
e
fi
nve
r
s
e
i
mage
sf
or e
ac
ht
i
me s
e
r
i
e
s ofl
e
ngt
h n. Combi
ni
ng t
hi
sf
ac
tand Pr
opos
i
t
i
on 5.1
c
ons
t
i
t
ut
e
st
het
he
or
e
t
i
c
albas
ef
ort
heus
eofECA ofr
ul
e 30 asa r
andom numbe
r
.
ge
ne
r
at
or
5.2 Thes
ubc
l
as
sI.2
He
r
ear
eonl
y4ECAswi
t
ht
her
ul
enumbe
r
sof94,122,126and184.Fore
ac
hof
e
t
he
mt
hee
vol
ut
i
onar
ymappi
ngf i
ss
ur
j
e
c
t
i
ve,butt
hemappi
ngfi
snots
ur
j
e
c
t
i
ve.I
ti
s
[7,8]
knownt
hatt
hef
or
me
rt
woECAshavenonr
e
gul
arl
i
mi
t
i
ngl
anguage
s
.Fort
hef
or
me
r
e
3ECAs
,t
hepr
oofoft
he
i
rf t
obes
ur
j
e
c
t
i
ver
e
l
i
e
sont
hef
ac
tt
hatt
he
yc
ans
i
mul
at
et
he
e
ECA ofr
ul
e90whos
ef i
ss
ur
j
e
c
t
i
ve.Thi
sc
anbes
i
mpl
ys
hownbyt
aki
ngt
hebl
oc
ks00
[1]
and11andt
het
i
mes
t
e
p2andi
somi
t
t
e
d .
,wegi
Si
nc
et
het
r
e
at
me
ntofECA ofr
ul
e184i
smor
edi
f
f
i
c
ul
t
vei
t
spr
oofbe
l
ow.
e
ss
ur
j
e
c
t
i
ve.
Pr
opos
i
t
i
on5.3.Thee
vol
ut
i
onar
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ngf184 i
e
.I
Pr
oof
ti
se
nought
opr
ovet
hatf
ore
ac
hxt
he
r
ee
xi
s
t
sones
t
r
i
ngas
uc
ht
hatf184(a)=x.
Si
nc
et
hel
anguageoft
i
mes
e
r
i
e
ss
at
i
s
f
i
e
st
hatf
ac
t
or
i
alpr
ope
r
t
y(2),wec
anr
e
s
t
r
i
c
t
:
ourat
t
e
nt
i
ont
ot
hef
ol
l
owi
ngt
ypeofs
t
r
i
ngs
x= 1m10n11m20n2...1mk0nk,
(9)
.
whe
r
eal
lofmi,niar
epos
i
t
i
vei
nt
ge
r
s
Thel
oc
alr
ul
eofECA ofr
ul
e184i
s
000,001,010,110→ 0;011,100,101,111→ 1.
I
ti
sus
e
f
ult
oc
oni
s
de
rt
hee
vol
ut
i
onf
i
gur
east
hatus
e
di
nExampl
e5.2.Put
t
i
ngt
he
t
i
mes
e
r
i
e
sxof(9)att
hes
i
t
e0f
r
om t
opt
obot
t
om,t
het
i
mes
e
r
i
e
s
b= 0m1- 10n1(10m2- 1)...(10mk- 1)0nk
att
hes
i
t
e-1,andt
het
i
mes
e
r
i
e
s
c= 1m1- 1(01n1- 1)1m2(01n2- 1)...1mk(01nk- 1),
)t
att
hes
i
t
e1s
i
mi
l
ar
l
y,t
he
nus
i
ngt
heope
r
at
i
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260
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28
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33
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78
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134
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