We work in the setting of linear skew constacyclic codes over a commutative base ring $S$. 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 isomorphisms between their ambient rings which extend some automorphism $τ$ of $S$ that commutes with $σ$ must have degree one, when those rings are not associative. In the process we determine isomorphisms between their nonassociative ambient rings, the Petit rings $S[t;σ]/S[t;σ](t^n-a)$, 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 $τ\in {\rm Aut}(S)$ that commute with $σ$, and lead to tighter classifications.

翻译：我们在交换基环$S$上的线性斜约束循环码框架中展开研究。我们证明了Ou-azzou等人引入的$(n,σ)$-等距与$(n,σ)$-等价概念在长度为$n$的大多数斜$(σ,a)$-约束循环码中是一致的。为证明这一结论，我们表明：当这些环境环非结合时，所有保持汉明权重的同构（该同构可扩展为与$σ$可交换的环$S$的自同构$τ$）其度数必然为1。在此过程中，我们确定了非结合环境环——佩蒂环$S[t;σ]/S[t;σ](t^n-a)$（该环生成斜约束循环码）之间的同构关系。由此，我们提出了斜约束循环码等价与等距的新定义，这些定义精确刻画了所有扩展自与$σ$可交换的自同构$τ∈{\rm Aut}(S)$的、保持环境环汉明权重的同构，并导致更严格的分类结果。