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=n1ps,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 ⊕b∈IS(rb) 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.
Highlights
The class of constacyclic codes plays an important role in coding theory and has been a primary area of study
Constacylic codes of arbitrary length N over a finite ring R are identified with ideals of the polynomials ring R[x]/ < xN − λ >
Motivated by the above cited studies, the main objective of this paper is to extend the approach of Han et al [15] and to obtain unique polynomial representations of constacyclic codes of any finite length N over R with arbitrary invariants p, r, and k
Summary
The class of constacyclic codes plays an important role in coding theory and has been a primary area of study (see [1,2,3,4,5,6,7,8,9]). The class of finite chain rings has been extensively used as the alphabet of constacyclic codes [8,13,14,15,16,17,18,19,20,21,22,23,24]. Motivated by the above cited studies, the main objective of this paper is to extend the approach of Han et al [15] and to obtain unique polynomial representations of constacyclic codes of any finite length N over R with arbitrary invariants p, r, and k.
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