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. 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