Skew-constacyclic codes over finite chain rings

Abstract

Skew polynomial rings over finite fields and over Galois rings have been used to study codes. In this work, we extend this concept to R(pm,e):= Fpm + uFpm + ⋅⋅⋅ + ue-1Fpm, a finite chain ring of prime characteristic p. The Gray images of codes over this ring are also studied. Given a unit λ∈ R(pm,e), properties of free skew-constacyclic codes are established corresponding to λ. When λ2=1, the generators of Euclidean and Hermitian duals of such codes are determined together with necessary and sufficient conditions for them to be Euclidean and Hermitian self-dual. Of more interest are codes over the ring R(pm,2):=Fpm+uFpm. The structure of all skew-constacyclic codes is completely determined. This allows us to express generators of Euclidean and Hermitian dual codes of skew-cyclic and skew-negacyclic codes in terms of the generators of the original codes. An illustration of all skew cyclic codes of length 2 over R(3,2) and their Euclidean and Hermitian duals is also provided. The Gray map is introduced for R(pm,e) to link codes over this ring and over its residue field. We prove that the Gray image of an (1-ue-1)-constacyclic code over R(pm,e) is a distance-invariant quasi-cyclic code over its residue field. When the length of codes is not divisible by p, the Gray images of a cyclic code and an (1+ue-1)-constacyclic code are permutatively equivalent to quasi-cyclic codes over its residue field. Finally, we give descriptions concerning Gray images of some skewconstacyclic codes over R(pm,e).