Constacyclic codes over finite chain rings of characteristic p

Abstract

Let R be a finite commutative chain ring of characteristic p with invariants p, r, and k. In this paper, we study λ-constacyclic codes of an arbitrary length N over R, where λ is a unit of R. We first reduce this to investigate constacyclic codes of length ps (N = n1 ps, p ∤ n1) over a certain finite chain ring CR(uk, rb) of characteristic p, which is an extension of R. Then we use discrete Fourier transform (DFT) to construct an isomorphism γ between R[x]/ < xN − λ > and a direct sum (Formula presented) of certain local rings, where I is the complete set of representatives of p-cyclotomic cosets modulo n1. By this isomorphism, all codes over R and their dual codes are obtained from the ideals of S(rb). In addition, we determine explicitly the inverse of γ so that the unique polynomial representations of λ-constacyclic codes may be calculated. Finally, for k = 2 the exact number of such codes is provided.