In this paper we generalize the notion of $n$-equivalence relation introduced by Chen et al. in \cite{Chen2014} to classify constacyclic codes of length $n$ over a finite field $\mathbb{F}_q$, where $q=p^r$ is a prime power, to the case of skew constacyclic codes without derivation. We call this relation $(n,\sigma)$-equivalence relation, where $n$ is the length of the code and $ \sigma$ is an automorphism of the finite field. We compute the number of $(n,\sigma)$-equivalence classes, and we give conditions on $ \lambda$ and $\mu$ for which $(\sigma, \lambda)$-constacyclic codes and $(\sigma, \lambda)$-constacyclic codes are equivalent with respect to our $(n,\sigma)$-equivalence relation. Under some conditions on $n$ and $q$ we prove that skew constacyclic codes are equivalent to cyclic codes. We also prove that when $q$ is even and $\sigma$ is the Frobenius autmorphism, skew constacyclic codes of length $n$ are equivalent to cyclic codes when $\gcd(n,r)=1$. Finally we give some examples as applications of the theory developed here.

翻译：本文推广了Chen等人在文献\cite{Chen2014}中提出的$n$-等价关系概念，该概念原本用于分类有限域$\mathbb{F}_q$（其中$q=p^r$为素数幂）上长度为$n$的常循环码，我们将其推广至无导数的偏斜常循环码情形。我们将此关系称为$(n,\sigma)$-等价关系，其中$n$为码长，$\sigma$为有限域的自同构。我们计算了$(n,\sigma)$-等价类的数目，并给出了$\lambda$和$\mu$的条件，使得$(\sigma,\lambda)$-常循环码与$(\sigma,\mu)$-常循环码关于$(n,\sigma)$-等价关系等价。在$n$和$q$满足一定条件时，我们证明了偏斜常循环码等价于循环码。此外，当$q$为偶数且$\sigma$为Frobenius自同构时，我们证明当$\gcd(n,r)=1$时，长度为$n$的偏斜常循环码等价于循环码。最后，我们给出一些算例以展示本文所发展理论的应用。