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In this paper we generalize the notion of $n$-equivalence relation introducedby 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 caseof 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 $q$ weprove that skew constacyclic codes are equivalent to cyclic codes. We alsoprove that when $q$ is even and $\sigma$ is the Frobenius autmorphism, skewconstacyclic codes of length $n$ are equivalent to cyclic codes when$\gcd(n,r)=1$. Finally we give some examples as applications of the theorydeveloped here.

On $(n,\sigma)-$equivalence relation between skew constacyclic codes