1-Generator quasi-cyclic and generalized quasi-cyclic codes over the ring $\frac{{{\mathbb {Z}_4}[u]}}{{\left\langle {{u^2} - 1}\right\rangle }}$

Abstract

In this paper, we study 1-generator quasi-cyclic and generalized quasi-cyclic codes over the ring $R=\frac{{{\mathbb {Z}_4}[u]}}{{\left\langle {{u^2} - 1} \right\rangle }}$. We determine the structure of the generators and the minimal generating sets of 1-generator QC and GQC codes. We also give a lower bound for the minimum distance of free 1-generator quasi-cyclic and generalized quasi-cyclic codes over this ring, respectively. Furthermore, some new $\mathbb {Z}_4$-linear codes via the Gray map which have better parameters than the best known $\mathbb {Z}_4$-linear codes are presented.