When isometry and equivalence for skew constacyclic codes coincide

TL;DR

This paper investigates when isometry and equivalence notions for skew constacyclic codes over rings coincide, providing new definitions and classifications based on ring isomorphisms that preserve Hamming weight.

Contribution

It establishes conditions under which isometry and equivalence coincide for skew constacyclic codes and introduces refined definitions capturing all Hamming-weight preserving isomorphisms.

Findings

Most skew $(\sigma,a)$-constacyclic codes have coinciding isometry and equivalence notions.

All Hamming-weight preserving isomorphisms extend automorphisms $ au$ of $S$ that commute with $\sigma$ and have degree one.

New definitions of code equivalence and isometry lead to tighter classification of skew constacyclic codes.

Abstract

We work in the setting of linear skew constacyclic codes over a commutative base ring $S$. We show that the notions of $(n,\sigma)$-isometry and $(n,\sigma)$-equivalence introduced by Ou-azzou et al coincide for most skew $(\sigma,a)$-constacyclic codes of length $n$. To prove this, we show that all Hamming-weight preserving isomorphisms between their ambient rings which extend some automorphism $\tau$ of $S$ that commutes with $\sigma$ must have degree one, when those rings are not associative. In the process we determine isomorphisms between their nonassociative ambient rings, the Petit rings $S[t;\sigma]/S[t;\sigma](t^n-a)$, which give rise to skew constacyclic codes. As a consequence, we propose new definitions of equivalence and isometry of skew constacyclic codes that exactly capture all Hamming-weight preserving isomorphisms between the ambient rings of skew constacyclic codes which extend $\tau\in {\rm Aut}(S)$ that commute with $\sigma$, and lead to tighter classifications.