On Constacyclic codes over Zpm - IEEE Xplore

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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)

REFERENCES Pramod Kanwar and Sergio R. L6pez-Permouth. cyclic codes over the integer modulo pm Finite field and their applications, 3:334-352, 1997. A.R. Calderbank, Pramod, and N.1.A. Sloane. Modular and p-adic codes. Designs, codes and Cryptography,

6:21-35, 1995.

Finite field and their applications,

16:243-253, 2010.

Shixin Zhu and Xiaoshan Kai. A class of constacyclic codes over IZpm.

[4]

Xiaoshan Kai, Shixin Zhu, and Yongsheng Tang. Some constacyclic self­ dual codes over integers modulo 2m. Finite field and their applications,

[5]

Yonglin Cao. On constacyclic codes over finite chain rings.

18:258-270, 2012.

and their applications,

[6]

Hai Q. Dinh. of Algebra,

[7]

[9]

Constacyclic codes of length pS

324:940-950, 2010.

Bernard R. McDonald.

1974. [8] W. Edwin

24:124-135, 2013.

Finite field

over IFpm +ulFpm. Journal

Finite Rings with Identity.

Dekker, New York,

Clark and Joseph J Liang. Enumeration of finite commutative chain rings. Journal of Algebra, 27:445-453, 1973. G.H. Pramod Norton and A. Salagean. On the structure of linear and cyclic codes over a finite chain ring. AAECC, 10:489-506, 2000.