On Quasi-cyclic Codes over Integer Residue Rings

Abstract

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 ${\mathbb{Z}}_{p_i^{e_i}}$, $k = \prod_{i=1}^s p_i^{e_i}$, p _i a prime. Let T be the standard shift operator. A linear code $\mathcal{C}$ over a ring R is called an l-quasi-cyclic code if $T^l(c) \in \mathcal{C}$, whenever $c\in \mathcal{C}$. It is shown that if (m, q) = 1, q = p ^r, p a prime, then an l-quasi-cyclic code of length lm over ℤ _q is a direct product of quasi-cylcic codes over some Galois extension rings of ℤ _q . We have discussed about the structure of the generator of a 1-generator l-quasi-cyclic code of length lm over ℤ _q . A method to obtain quasi-cyclic codes over ℤ _q , which are free modules over ℤ _q , has been discussed.