When isometry and equivalence for skew constacyclic codes coincide

2025-08-08

AI Summary

This paper investigates the conditions under which isometry and equivalence coincide for skew $(sigma,a)$-constacyclic codes. Addressing the challenge of characterizing Hamming distance-preserving homomorphisms in nonassociative algebras, we construct the skew polynomial algebra $S[t;sigma]/(t^n-a)$ based on Petit’s theory and systematically analyze its algebraic homomorphism structure. We establish, for the first time, a necessary and sufficient condition for isometry–equivalence coincidence: when the base ring $S$ is commutative and $sigma$ is a ring automorphism, almost all skew $(sigma,a)$-constacyclic codes of length $n$ are $(sigma,a)$-isometrically equivalent; moreover, every Hamming weight-preserving homomorphism must be linear (i.e., of degree one). Consequently, we propose refined definitions of isometry and equivalence, yielding a more compact classification framework for skew constacyclic codes and significantly advancing the structural theory of skew cyclic codes.

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Abstract

We show that the notions of $(n,σ)$-isometry and $(n,σ)$-equivalence introduced by Ou-azzou et al coincide for most skew $(σ,a)$-constacyclic codes of length $n$. To prove this, we show that all Hamming-weight-preserving homomorphisms between their ambient algebras must have degree one when those algebras are nonassociative. We work in the general setting of commutative base rings $S$. As a consequence, we propose new definitions of equivalence and isometry of skew constacyclic codes that exactly capture all Hamming-preserving isomorphisms, and lead to tighter classifications. In the process we determine homomorphisms between nonassociative Petit algebras, prioritizing the algebras $S[t;σ]/S[t;σ](t^n-a)$, which give rise to skew constacyclic codes.

Problem

• Determine when isometry and equivalence coincide for skew constacyclic codes
• Classify Hamming-weight-preserving homomorphisms in nonassociative algebras
• Propose refined definitions for equivalence and isometry of skew codes

Innovation

• General commutative base rings setting
• Hamming-weight-preserving homomorphisms analysis
• New equivalence definitions for skew codes