Consta-abelian codes over galois rings in the dft domain

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 Consta-Abelian codes over Galois rings that are closed under two kinds of monomials.

I. INTRODUCTION AND PRELIMINARIES

For a primep, Galois ring &a.,' = 2,.[ x ] / ~ ( x ) , where

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 =

nlL',

^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 ) =

$'k(gtmh),$mh))S'i2h), where g:Zk) E and

where

pk,

an element of order r k belongs to the cyclic sub- group 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 j ) ) also belon to the code for all values of j , where r i ( c i e j ) =

(n,=, p:')

^{ciaj and kx} is such that i x + j , = mxk, + ( i x $ j x ) where i @Ij 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 =

ctx,

^{CYA =}

si"

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

,...,

A n . - l ) E R;a,lm of a'= ( u o , . .

.

, a n - 1 ) E RFa.1 is defined as Aj = ( n ; ~ ~ y ~ c ~ > ~ ~ ) ai 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 proper- ties 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 au- tomorphism of Rpa,lm that fixes Rp.1 Rpa,lm. Moreover, for j E I,, u k ( A j ) = Ai, where i = [i,-~,i,-~,...,i01 with ix =

(w + q k j , ) modulo mx for all X = 0 , 1 , . . . , r - 1 (Conjugate symmetry property). Let 4: denote a mapping from Im, to Imx which maps j x to (* + q k j x ) . Let

ak

denote the mapping i = [ i r - l , i r - 2 , .

. . ,

io] to @(i) =

r ~ ~ - l ( ~ , - ~ ) , ~ ~ - 2 ( ~ ~ - z ) , . . . , ~ ~ ( ~ ~ ) l . Foreveryj E I,, theset

= { @ ' ( j ) , iP 2 ( j ) ,

. .

.

,

a e i - ' ( j ) } where ej is the smallest integer such that C P " j ( j ) = j, is called the cyclotomic coset

7 3

'This work was partly supported by CSIR, India, through Re- search Grant (22(0298)/99/EMR-I1) to B.S.Rajan.

modulo n containing rj1

.

A linear code C over Rpa,l is consta- Abelian iff it satisfies the conjugate-symmetry property and the set Cj = { A j

I

15 E C} = p"j &a,lei, an ideal of the subring

q a , l e j where 0

5

qj

5

a.

In this paper, we study P-constacyclic codes which are also invariant under the monomid-permutations d b ) U b 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 as- sociated permutation

d b )

= ( r f ) ,

. . . ,T??~)

is given by r y ) ( a i ) = n L z : $ ' k

(

^{gmh b'lik .ui)} and (ii) for ns =

r L i

^mx,

Qns takes i

+

( i

+

n , ) modulo n and the associated permu- tation A<'> = (r:",.

. . ,

r::;) is given by ai) = ai for 0

5

i

<

n,+l

-

ns and for n,+l

-

ns

5

ⁱ

<

n, ai) = n L z : + h

(

^{gmr, ai), where 6t)} denotes the

"carry value" in the k-th radix-component due t o addition [i

+

^{n , l .}

11. TDFT DOMAIN CHARACTERIZATION

Definition 1 (a) For any j E I n such that [jl=[Ol 0 , .

. .

, 0 , j,i#

O , j p - l ,

. . . ,

jol and p 2 h

>

s 2 0 , let the set J @ , " ) ( j ) be

X X E I m A ; X = h-1, h - 2 , .

. . ,

^S. (ai) For every xs), 1 , . . .

,

r-1,

let q 5 : I b A denote the mapping from I m x to Im, that maps j x to

+

b;"jA) and let @ k , b be the map which maps

rjl

Theorem 1 A length n = mr-1mr-2..

.

mo P-constacyclic code with Cj = p"JRpa,lej for any j E In is (i) r(*'Ub- invariant a# Ci = p"j&-,rei when some element of [i] i s of the form @ k 3 b ( [ j l ) for some k. (ii) r<"Qn,-invariant h ifl Ci = pqJRpa,lei when j 2 n,+l and some element of [il be- longs to ~ ( ~ - ' l ~ ) ( j ) ,

= 1 for all A, Theorem 1 char- acterizes (i) Quasicyclic-Abelian codes and (ii) &invariant Abelian codes. Given the TDFT characterization of a

p-

constacyclic code, we obtain the characterization of its dual code (w.r.t normal inner product).

ih+6!"'

{Yo,-

, O l j p 1 . . . , j h , ~ h - - 1 , X h - 2 , . . .

,

~ , , j ~ - ~ , . .

. ,jol}

f o r all

(

^{( b y 1;,' )h x}

k,b,-l

to aklb(rji) =

r4T-l (jr-l),4:5-z(jT-2), . .

. , & 7 b o ( j o ) i .

h

For the special case of

REFERENCES

B.R.McDonald, "Finite Rings with Identity," Marcel Dekker Inc., 1974, New York.

G.Hughes, "Structure theorems for Group Ring Codes with an application to Self-Dual Codes," Des. Codes. Cryptogr., vol. 24, pp. 5-14, Sept-2001.

B.Sundar Rajan and M.U.Siddiqi, "A Generalized DFT for Abelian Codes over Zm," IEEE Trans. Info. Theory, vol. 40, no. 6, pp. 2082-2090, Nov-1994.

Serdar Boztas, "Constacyclic Codes and Constacyclic DFTs."

Proc. of ISIT-1998, pp. 235, Cambridge, MA, USA,

0-7803-7728-1/03/$17.00 02003 IEEE.

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