On Quasi-cyclic Codes over Integer Residue Rings
In this paper we consider some properties of quasi-cyclic codes over the integer residue rings. A quasi-cyclic code over ℤ k , the ring of integers modulo k, reduces to a direct product of quasi-cyclic codes over \({\mathbb{Z}}_{p_i^{e_i}}\) , \(k = \prod
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Centre for Development of Advanced Computing, Noida, India [email protected] Department of Mathematics, Jamia Millia Islamia, New Delhi, India [email protected]
Abstract. In this paper we consider some properties of quasi-cyclic codes over the integer residue rings. A quasi-cyclic code over Zk , the ring of integers modulo k, reduces to a direct product of quasi-cyclic codes over Zpei , k = si=1 pei i , pi a prime. Let T be the standard shift i operator. A linear code C over a ring R is called an l-quasi-cyclic code l if T (c) ∈ C, whenever c ∈ C. It is shown that if (m, q) = 1, q = pr , p a prime, then an l-quasi-cyclic code of length lm over Zq is a direct product of quasi-cylcic codes over some Galois extension rings of Zq . We have discussed about the structure of the generator of a 1-generator lquasi-cyclic code of length lm over Zq . A method to obtain quasi-cyclic codes over Zq , which are free modules over Zq , has been discussed.
Keywords: Quasi-cyclic codes, circulant matrices, Galois rings, Hensel’s lift.
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Introduction
Quasi-cyclic codes form a remarkable generalization of cyclic codes [1], [5], [18]. They are asymptotically good as they meet a modified version of GilbertVarshamov bound [10]. They are closely linked to convolutional codes. Recently there has been a great interest for their applications in studying Turbo codes [19] and many Low-Density Parity Check (LDPC) codes [6]. It is well known that Turbo codes and LDPC codes have been proved to be capacity approaching codes. In recent years, there has been a lot of interest in codes over finite rings since the revelation in 1994, in a breakthrough paper by Hammons et al. [7], that some non linear binary codes with very good parameters are actually binary images under a certain map, of some linear codes over Z4 , the ring of integers modulo 4. Cyclic codes over some rings have been studied by a number of authors [2], [3], [15], [16], however, there has been a limited study on quasi-cyclic codes over rings [1], [12]. Let R be finite commutative ring with identity. A linear code C of length n over R is a submodule of the R-module Rn . Such a submodule is not necessarily S. Bozta¸s and H.F. Lu (Eds.): AAECC 2007, LNCS 4851, pp. 330–336, 2007. c Springer-Verlag Berlin Heidelberg 2007
On Quasi-cyclic Codes over Integer Residue Rings
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a free module. Let T be the standard shift operator. A linear code C over R is called a quasi-cyclic code of index l (or an l-quasi-cyclic code ) if T l (c) ∈ C whenever c ∈ C. For l = 1, a quasi-cyclic code is simply a cyclic code. We assume that l divides the code length n and n = lm, for some positive integer m. Up to equivalence, the generator matrix of an l-quasi-cyclic code of length n = lm can be expressed as a block matrix of m × m circulant matrices [4]. It is well known that a quasi-cyclic code of length n = lm over R can be regarded as l R[x] [1], [12]. If this module is generated by a an xR[x] m −1 -submodule of m x −1 l single element g (x ) ∈ xR[x] , then we say that the code
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