ISlT 2003, Yokohama,Japan, June 29 -July 4, 2003
Consta-Abelian codes over Galois rings in the DFT domain
Kiran. T, and
B. Sundar Rajan
ECE Dept., Indian Institute of Science, Bangalore 560 012, India
{kirant(Oprotocol., bsrajanQ}ece.iisc.ernet.in
Abstract - Using Twisted-DFT, we characterize modulo n containing rj1 . A linear code C over Rpa,lis constaConsta-Abelian codes over Galois rings that are closed Abelian iff it satisfies the conjugate-symmetry property and
the set Cj = {Aj I 15 E C} = p"j &a,lei, an ideal of the subring
under two kinds of monomials.
q a , l e j where 0 5 qj 5 a.
I. INTRODUCTION
AND PRELIMINARIES
For a primep, Galois ring &a.,' = 2,. [ x ] / ~ ( xwhere
) , 4(z) is a
monic irreducible polynomial of degree 1 over Zp[2] [l].If G =
C,-1 @ . . . @ CO,
is an Abelian group of order n =
mx,
the twisted group ring R:a,lG [2] is defined by imposing in the
Rp,l-module
G the following multiplication on each of
the generators g(mh)of the cyclic subgroup c k : Simh).g { m h ) =
nlL',
and
$'k(gtmh),$mh))S'i2h), where g:Zk) E
In this paper, we study P-constacyclic codes which are
also invariant under the monomid-permutations db)Ub and
r<'>Qn. where: (i) ub is defined by, [il = [i, - I , . . . , i o 1 -$
[ b , - l i , - l , . . . ,boio] for b = [ L 1. .,,bel
. E I, such that
gcd(b,,mx) = 1 and T A I (b,' - 1 ) for all A. The associated permutation d b )= ( r f ) ,. . .
is given by
ry)(ai)= nLz:$'k
(n,=,
(w+
si"
from Im, to Imx
which maps j x to
ak denote
the mapping i = [ i r - l , i r - 2 , .
+qkjx).
(
ih+6!"'
gmr,
nLz:+h
ai), where 6t) denotes the
11. TDFT
DOMAIN CHARACTERIZATION
Definition 1 (a) For any j E I n such that [jl=[Ol0 , . . . ,0, j,i#
O , j p - l , . . . ,jol and p 2 h > s 2 0 , let the set J @ , " ) ( j )be
{Yo,- , O l j p 1 . . . , j h , ~ h - - 1 , X h - 2 , . . . ,~ , , j ~ -. .~,jol}
, . f o r all
X X E I m A ;X = h-1, h - 2 , . . . , S. (ai) For every xs),1 , . . . , r-1,
let
denote the mapping from I m x to Im, that maps j x to
(
q5:IbA
(1b; y, '
)h x
+ b;"jA)
k,b,-l
to aklb(rji)= r4T-l
and let
@k,b
be the map which maps
rjl
(jr-l),4:5-z(jT-2),
.. . ,&7bo(jo)i.
Theorem 1 A length n = mr-1mr-2.. . mo P-constacyclic
code with Cj = p"JRpa,lejfor any j E In is (i) r(*'Ubinvariant a# Ci = p"j&-,rei when some element of [i] i s of
) some k. (ii) r<"Qn,-invariant ifl
the form @ k 3 b ( [ j lfor
Ci = pqJRpa,lei when j 2 n,+l and some element of [il be~longs
) to
h
h
~ ( ~ - ' l ~ ) ( j ) ,
For the special case of
= 1 for all A, Theorem 1 characterizes (i) Quasicyclic-Abelian codes and (ii) &invariant
Abelian codes. Given the TDFT characterization of a pconstacyclic code, we obtain the characterization of its dual
code (w.r.t normal inner product).
Let
. . ,io]
to @(i)
=
E I,, theset
= { @ ' ( j ) , i P 2 ( j ) , . . . , a e i - ' ( j ) }where ej is the smallest
integer such that C P " j ( j ) = j,is called the cyclotomic coset
r ~ ~ - l ( ~ , -~ ~ -) 2, ( ~ ~ - z ) , . . . , ~ ~ (Foreveryj
~~)l.
'This work was partly supported by CSIR, India, through Research Grant (22(0298)/99/EMR-I1) to B.S.Rajan.
0-7803-7728-1/03/$17.00
02003 IEEE.
rLimx,
+
4: denote a mapping
(*
.ui) and (ii) for ns =
"carry value" in the k-th radix-component due t o addition
[i n , l .
q k j , ) modulo mx for all X = 0 , 1 , .. . , r - 1
(Conjugate symmetry property). Let
gmh
b'lik
+
ai) =
ix =
(
Qns takes i + ( i
n,) modulo n and the associated permutation A<'> = (r:",. . . ,r::;) is given by ai) = ai
for 0 5 i < n,+l - ns and for n,+l - ns 5 i < n,
where pk, an element of order r k belongs to the cyclic subgroup of size p' - 1 in R;a,l. For n permutations TO,.. . , Kn-1
of &a,'
and a permutation r of In = { O , l , . ~ . ,n 1) a
code C over Rpa,l is said to be rr-invariant if rr(4 =
( r o ( c ~ ( o ) ) , T i ( c T ( i ) ) ,. . . , T n - l ( ~ ~ ( ~ - l )E) ) C for all 3 =
( G J , C ~ , . . . , C ~ - ~E) C.
Consta-Abelian codes are ideals in
the twisted group ring R:a,,G. For p = @ , - I , . . . ,PO) these
are also called B-constacyclic codes. For every codeword
Z = (Q, . . . ,C n - 1 ) in the P-constacyclic code, the monomial))
permuted vector rr(4 = ( r o ( c o ~ j ).,. . , r , + - l ( q n - l ) e jalso
belon to the code for all values of j , where r i ( c i e j ) =
7
3
p:') ciaj and kx is such that i x + j , = mxk, + ( i x $ j x )
where i @I j denotes the mixed-radix addition with mx as
radixes 131. Let <A be a primitive m x r x - t h root of unity in
the cyclic group of order p" - 1 in the extension ring R p n , l m
and "yx =
CYA =
be mx-th root of PA and mx-th root
of unity-respectively for all X = 0,1,. . . , r - 1. The TDFT
vector A = (A0,...,An.-l) E R;a,lmof a'= ( u o , . .. , a n - 1 ) E
RFa.1 is defined as Aj =
( n ; ~ ~ y ~ aci ~where
> ~
i = [ i , . - l , . . . , i o l and j = [ j , - l , . . . , j o l are mixed-radix
representations of i , j. The TDFT satisfies all the properties mentioned in [4] for a constacyclic DFT. Let (TO be the
F'robenius automorphism of I t p a , l m then U = U; is an auRpa,lm. Moreover,
tomorphism of Rpa,lm that fixes Rp.1
for j E I,, u k ( A j ) = Ai, where i = [i,-~,i,-~,...,i01 with
ctx,
,T??~)
332
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