On quantum codes derived from quasi-cyclic codes over a non-chain ring

This paper presents a study on the structure of 1-generator quasi-cyclic (QC) codes over the non-chain ring \(R=\mathbb {F}_{q}+u\mathbb {F}_{q}+v\mathbb {F}_{q}+uv\mathbb {F}_{q}\) $R={\mathbb{F}}_q+u{\mathbb{F}}_q+v{\mathbb{F}}_q+uv{\mathbb{F}}_q$ , where \(u^2=v^2=0,~ uv=vu\) $u^2=v^2=0,uv=vu$ , and \(\mathbb {F}_q\) ${\mathbb{F}}_q$ is a finite field of cardinality \(q=p^r\) $q=p^r$ ; p is a prime. A minimal spanning set and size of these codes are determined. A sufficient condition for 1-generator QC codes over R to be free is given. BCH-type bounds on the minimum distance of free QC codes over R are also presented. Some optimal linear codes over \(\mathbb {F}_q\) ${\mathbb{F}}_q$ are obtained as the Gray images of quasi-cyclic codes over R. Some characterizations of the Gray images of QC codes over R in \(\mathbb {F}_q\) ${\mathbb{F}}_q$ and \(\mathbb {F}_q+u\mathbb {F}_q~(u^2=0)\) ${\mathbb{F}}_q+u{\mathbb{F}}_q(u^2=0)$ are done. As an application, we consider self-orthogonal subcodes of the Gray images of QC codes over R to obtain new and better quantum codes than those are available in the literature.