On isometry and equivalence of skew constacyclic codes

In this paper we generalize the notion of n-isometry and n-equivalence relation introduced by Chen et al. in [13,12] to classify constacyclic codes of length n over a finite field Fq, where q = pr is a prime power, to the case of skew constacyclic codes without derivation. We call these relations respectively (n, sigma)-equivalence and (n, sigma)-isometric 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 and (n, sigma)-isometric classes, and we give conditions on lambda and mu for which (sigma, lambda)-constacyclic codes and (sigma, mu)-constacyclic codes are equivalent. Under some conditions on nand q we prove that skew constacyclic codes are equivalent to cyclic codes by using properties of our equivalence relation introduced. We also prove that when q is even and sigma is the Frobenius automorphism, 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. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.