Skew polycyclic over finite chain rings associated to trinomials

2026-05-04

AI Summary

This study addresses the classification of skew polycyclic codes over finite chain rings, defined by central trinomials, under Hamming equivalence in the noncommutative setting. By introducing a suitable equivalence relation on the defining polynomials, the general case is effectively reduced to the classification problem for codes with standard generator polynomial of the form $x^n - (x^\ell + 1)$. Leveraging tools such as the unit group decomposition of finite chain rings, Schur products, and group actions, the work establishes—for the first time in a noncommutative framework—a group-theoretic characterization of Hamming equivalence for these codes. This approach not only elucidates the algebraic structure of the equivalence classes but also yields an exact count of their number, thereby significantly simplifying the classification of skew polycyclic codes.

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.

Problem

• skew polycyclic codes
• finite chain rings
• trinomials
• Hamming equivalence
• non-commutative

Innovation

• skew polycyclic codes
• finite chain rings
• Hamming equivalence
• trinomials
• group-theoretic characterization