On skew cyclic codes over $ M_{2}(\mathbb{F}_{2}) $

Abstract

The algebraic structure of skew cyclic codes over $ M_{2} $($ \mathbb{F}_2 $), using the $ \mathbb{F}_4 $-cyclic algebra, is studied in this work. We determine that a skew cyclic code with a polynomial of minimum degree $ d(x) $ is a free code generated by $ d(x) $. According to our findings, skew cyclic codes of odd and even lengths are cyclic and $ 2 $-quasi-cyclic over $ M_{2}(\mathbb{F}_{2}) $, respectively. We provide the self-dual skew condition of Hermitian dual codes of skew cyclic codes. The generator polynomials of Euclidean dual codes are obtained. Furthermore, a spanning set of a double skew cyclic code over $ M_{2}(\mathbb{F}_{2}) $ is considered in this paper.

Keywords

• skew cyclic code
• finite chain ring
• skew polynomial rings
• gray map