On quasi-cyclic codes over $${\mathbb{Z}_q}$$

Abstract

Quasi-cyclic (QC) codes are a remarkable generalization of cyclic codes. Many QC codes have been shown to be best for their parameters. In this paper, some structural properties of QC codes over the prime power integer residue ring $${\mathbb{Z}_q}$$are considered. An l-QC code of length lm over $${\mathbb{Z}_q}$$is viewed both as in the conventional row circulant form and also as a $${\frac{\mathbb{Z}_q[x]}{\langle x^m-1 \rangle}}$$-submodule of $${\frac{GR(q,l)[x]}{\langle x^m-1 \rangle}}$$, where GR(q, l) is the Galois extension ring of degree l over $${\mathbb{Z}_q}$$. A necessary and sufficient condition for cyclic codes over Galois rings to be free is obtained and a BCH type bound for them is also given. A sufficient condition for 1-generator QC codes to be $${\mathbb{Z}_q}$$-free is given and a formula to evaluate their ranks is derived. Some distance bounds for 1-generator QC codes are also discussed. The duals of QC codes over $${\mathbb{Z}_q}$$are also briefly discussed.