Skew polycyclic over finite chain rings associated to trinomials

TL;DR

This paper investigates skew polycyclic codes over finite chain rings defined by central trinomials, focusing on Hamming equivalence and classifying codes via group-theoretic methods.

Contribution

It introduces an equivalence relation on defining trinomials, characterizes it using group theory, and simplifies the classification of these codes up to Hamming equivalence.

Findings

Established conditions for Hamming equivalence to a canonical trinomial case

Provided a group-theoretic characterization of the equivalence relation

Determined the size of the equivalence classes using the unit group decomposition

Abstract

This work studies skew polycyclic codes over finite chain rings defined by central trinomials. For this class of codes, we investigate Hamming equivalence in the non-commutative (skew) setting. We introduce an equivalence relation on the defining trinomials and demonstrate that it admits a group-theoretic characterization in terms of a group of binomials equipped with the Schur multiplication. We determine the conditions under which skew polycyclic codes are Hamming equivalent to those defined by the specific trinomial $x^n-(x^\ell+1)$. This reduces the classification problem for these codes, up to Hamming equivalence, to a canonical case. Finally, we determine the size of the corresponding equivalence class using the decomposition of the unit group of the underlying chain ring.