Skew-Quasi-Cyclic Codes over $M_l({\mathbb{F}}_q)[X,\theta ]$

Abstract

Skew-quasi-cyclic codes over a finite field are viewed as skew-cyclic codes on a noncommutative ring of matrices over a finite field. This point of view gives a new construction of skew-quasi-cyclic codes. Let ${\mathbb{F}}_q$ be the Galois field with $q$ elements and $\theta$ be an automorphism of ${\mathbb{F}}_q$. We propose an approach to consider the relationship between left ideals in $M_l({\mathbb{F}}_q)[X,\theta ]/(X^s–1)$ and skew-quasi-cyclic codes of length $ls$ and index $l$ over ${\mathbb{F}}_q$, under $\theta$, which we denote by $\theta$-SQC codes (or SQC codes for short when there is no ambiguity). We introduce the construction of SQC codes from the reversible divisors of $X^s–1$ in $M_l({\mathbb{F}}_q)[X,\theta ]$. In addition, we give an algorithm to search for the generator polynomials of general SQC codes.