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.