On the algebraic structure of quasi-cyclic codes of index 112$1\frac {1}{2}$

Abstract

In this paper, we study quasi-cyclic codes of index $1\frac {1}{2}$ and co-index 2m over $\mathbb {F}_{q}$ and their dual codes, where m is a positive integer, q is a power of an odd prime and $\gcd (m,q) = 1$. We characterize and determine the algebraic structure and the minimal generating set of quasi-cyclic codes of index $1\frac {1}{2}$ and co-index 2m over $\mathbb {F}_{q}$. We note that some optimal and good linear codes over $\mathbb {F}_{q}$ can be obtained from this class of codes. Furthermore, the algebraic structure of their dual codes is given.