On isometry and equivalence of constacyclic codes over finite chain rings

In this paper we generalize the notions of n-isometry and n-equivalence introduced by Chen et al. to classify constacyclic codes of length \(n\in \mathbb {N}\) $n\in \mathbb{N}$ over a finite field \(\mathbb {F}_q\) ${\mathbb{F}}_q$ , to the case of constacyclic codes over a finite chain ring R. We show that these notions define equivalence relations on \(R^\times \) $R^{\times }$ , give equivalent characterizations of these relations and investigate the link between them. We then compute the numbers of n-isometry and n-equivalence classes for each type of finite chain rings, as well as describe methods to find these classes. In doing so, the study of constacyclic codes on a finite chain ring R is reduced significantly, as instead of considering \(\lambda \) $\lambda$ -constacyclic codes for each unit \(\lambda \) $\lambda$ in \(R^\times \) $R^{\times }$ , it suffices to only consider units in a set of representatives of the equivalence classes.