Email: [email protected] ... Keywords-Chain ring, Constacyclic codes, Dual codes, Self ... The study of linear codes over finite rings has received much.
On Constacyclic codes over Zpm Mohammed Elhassani Charkani
Joel Kabore
Department of Mathematics
Department of Mathematics
Faculty of Sciences, Dhar-Mahraz
P.
1796,
O. Box
Faculty of Sciences, Dhar-Mahraz
Atlas-Fez, Morocco
P.
N
= pkn
with gcd (p, n)
= 1.
In this work, we give a simple and
N - (1 + Ap) > short proof that the quotient ring R[X]I < X is a principal ring. This allow us to study (1 + Ap) -constacyclic codes of arbitrary length and give a characterization of self-dual
(1 + Ap) -constacyclic
codes over Zpm.
A.
Constacyclic codes over finite chain ring A finite ring is called chain ring if its ideals are linearly
ordered by inclusion. The finite fields, the ring of integers modulo pm and the Galois rings are examples of chain rings. The following result is well-known for finite chain ring (see
[6]):
Keywords- Chain ring, Constacyclic codes, Dual codes, Self dual codes.
I.
INTRODUCTION
The study of linear codes over finite rings has received much attention in recent years. An important class of linear codes are constacyclic codes wich are generalizations of cyclic and negacyclic codes. These codes posses rich algebraic structures and can be efficiently encoded using shift registers. It's well
1. Let R be a finite ring, the following conditions are equivalent: 1) R is a local ring and the maximal ideal of R is principal, 2) R is a local principal ideal ring, 3) R is a chain ring.
Proposition
If we denote by
R[xJI < x n - 1 >.
Similarly, for a given unit A,
A-constacyclic codes are ideals of the ring R[xl/
< xn
-A
>.
When studying constacyclic codes over finite rings, many authors assume that the code length is relativy prime with the characteristic of the residue field of the ring. This assures that the polynomial xn
-
A will have no multiple factors;
in this case the codes are called simple root codes, else they are called repeated root codes. Kanwar and Lopez in Calderbank and Sloane in
[2],
[1],
and
gave the structure of simple root
be a (1 + >..p) -constacyclic code over 'lL pm, then Pk m-l'r+l h * Pk m-lr+l l.. _ pk C - < gI m-I, ···grpk m-lrh r+I ( r+I) ...
IV.
{
From previous lemma,
is self-dual if and only if:
li = pkm -Ii ' lr+i = pkm - lr+i , lr+i = pkm - lr+i
1 :s; i :s; 1 :s; i :s; 1 :s; i :s;
If
r, rI, rI·
pm.
ACKNOWLEDGMENT The authors would like to thank the reviewers for providing
[3]
and only if there is no quasi self-reciprocal polynomials in the D
Let C be a (1+2>..)-constacyclic code of length 28 = 22. 7 over 'lL25 [xl. In 'lL25 x [ l
Example
over the ring of integers modulo
[2]
then
xn - 1.
(1+>..p) -constacyclic codes of arbitrary (1+>..p) -constacyclic code
length and characterized self-dual
[1]
_
factors of
Kai in [3]. We studied
very helpful COlmnents and suggestions.
Ii = 2k-Im for 1 :s; i :s; r; if in addition all factors of xn - 1 are quasi self-reciprocal polynomials, 1 k lm >=< (xn 1)2k-1m>. C =< gt-m... g;2) If p > 2 and m is even, then Ii = pkm/2, but if m is odd, the equality Ii = pkm/2 is absurd, so C is self-dual if 1)
p = 2,
C
CONCLUSION
This work was based on the results of Cao in [5], Zhu and
.
m-(+.r1 h * h pkr+r, ( r+r, ) pk m-lr+rl > .
ftif f� >=< (x4 - 1)3 > .
1.
x7 - 1 = (x - 1)(x3 +6x2+5x - 1)(x3 +27x2+26x - 1)
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