Characterizations and Properties of Monic Principal Skew Codes over Rings

Abstract

Let $A$ be a ring with identity, $\sigma$ a ring endomorphism of $A$ that maps the identity to itself, $\delta$ a $\sigma$ -derivation of $A$ , and consider the skew-polynomial ring $A\left[X;\sigma ,\delta \right]$ . When $A$ is a finite field, a Galois ring, or a general ring, some fairly recent literature used $A\left[X;\sigma ,\delta \right]$ to construct new interesting codes (e.g., skew-cyclic and skew-constacyclic codes) that generalize their classical counterparts over finite fields (e.g., cyclic and constacyclic linear codes). This paper presents results concerning monic principal skew codes, called herein monic principal $\left(f,\sigma ,\delta \right)$ -codes, where $f\in A\left[X;\sigma ,\delta \right]$ is monic. We provide recursive formulas that compute the entries of both a generator matrix and a control matrix of such a code $\mathcal{C}$ . When $A$ is a finite commutative ring and $\sigma$ is a ring automorphism of $A$ , we also give recursive formulas for the entries of a parity-check matrix of $\mathcal{C}$ . Also, in this case, with $\delta =0$ , we present a characterization of monic principal $\sigma$ -codes whose dual codes are also monic principal $\sigma$ -codes, and we deduce a characterization of self-dual monic principal $\sigma$ -codes. Some corollaries concerning monic principal $\sigma$ -constacyclic codes are also given, and a good number of highlighting examples is provided.