On $\mathbb{F}_{p^m}\mathfrak{R}$-additive skew negacyclic codes

Abstract

Let $\mathfrak{R} = \mathbb{F}_{p^m} + u\mathbb{F}_{p^m}$ with $u^2 = 0$, where $p$ is an odd prime and $m$ is any positive integer. This article delves into the algebraic structure of skew negacyclic codes of length $2p^s$ over a finite field $\mathbb{F}_{p^m}$ and a finite chain ring $\mathfrak{R}$. The focus is on the classification and structural properties of these codes. Based on the different possible factorizations of $x^{2p^s} + 1$ over $\mathbb{F}_{p^m}$, a complete classification of the structural properties of skew negacyclic codes and their duals for length $2p^s$ over $\mathbb{F}_{p^m}$ and $\mathfrak{R}$ is provided. Furthermore, the algebraic structure of $\mathbb{F}_{p^m}\mathfrak{R}$-additive skew negacyclic codes with block length $(p^s, 2p^s)$ is discussed. The separability of $\mathbb{F}_{p^m}\mathfrak{R}$-additive skew negacyclic codes is also analyzed. To illustrate these results, several examples are presented, including the construction of Maximum Distance Separable (MDS) and near-MDS codes.