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 $\Fq,$ 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}中引入的$n$等价关系概念，该概念用于分类有限域$\Fq$（其中$q=p^r$为素数幂）上长度为$n$的常循环码，并将其推广至无导数的斜常循环码情形。我们将这种关系称为$(n,σ)$等价关系，其中$n$为码长，$\sigma$为有限域的自同构。我们计算了$(n,σ)$等价类的个数，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$(n,σ)$等价关系等价。在$n$和$q$的某些条件下，我们证明了斜常循环码等价于循环码。我们还证明，当$q$为偶数且$\sigma$为Frobenius自同构时，若$\gcd(n,r)=1$，则长度为$n$的斜常循环码等价于循环码。最后我们给出了一些例子作为所发展理论的应用。