On a class of constacyclic codes of length 4ps over 𝔽pm[u] 〈u3〉

Abstract

Let [Formula: see text] be a prime such that [Formula: see text], where [Formula: see text] is a positive integer. For any nonzero element [Formula: see text] of [Formula: see text], we determine the algebraic structure of all [Formula: see text]-constacyclic codes of length [Formula: see text] over the finite commutative chain ring [Formula: see text], where [Formula: see text] and [Formula: see text] is a positive integer. If the unit [Formula: see text] is a square, [Formula: see text], each [Formula: see text]-constacyclic code of length [Formula: see text] is expressed as a direct sum of an [Formula: see text]-constacyclic code and an [Formula: see text]- constacyclic code of length [Formula: see text]. In the main case that the unit [Formula: see text] is not a square, it is shown that any nonzero polynomial of degree at most [Formula: see text] over [Formula: see text] is invertible in the ambient ring [Formula: see text]. It is also proven that the ambient ring [Formula: see text] is a local ring with the unique maximal ideal [Formula: see text], where [Formula: see text]. Such [Formula: see text]-constacyclic codes are then classified into eight distinct types of ideals, and the detailed structures of ideals in each type are provided. Among other results, the number of codewords, and the dual of each [Formula: see text]-constacyclic code are obtained. The non-existence of self-dual and isodual [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text], when the unit [Formula: see text] is not a square, is likewise proved.