Negacyclic self-dual codes over finite chain rings

Abstract

In this article, we study negacyclic self-dual codes of length n over a finite chain ring R when the characteristic p of the residue fieldR and the length n are relatively prime. We give necessary and sufficient conditions for the existence of (nontrivial) negacyclic selfdual codes over a finite chain ring. As an application, we construct negacyclic MDR self-dual codes over GR(p t , m) of length p m + 1.

Left dihedral codes over Galois rings GR(p2,m)

“…, c n−1 ) ∈ C, and C is said to be cyclic (negacyclic) when λ = 1 (λ = −1). Readers are referred to [17], [20], [21], [25], [27], [32] and [34] for results on linear codes, cyclic codes and constacyclic codes over finite chain rings. Moreover, various decoding schemes for codes over Galois rings have been considered in [11][12][13].…”

Primitive Idempotents and Constacyclic Codes over Finite Chain Rings

“…Since X n + 1 = (X 2n − 1)/(X n − 1), then X n + 1 can be factored uniquely into monic irreducible pairwise coprime polynomials as follows (see [7]):…”

“…This ensures that the polynomial X n − λ have no multiple factor; in this case the codes are called simple root constacyclic codes, else they are called repeated root constacyclic codes. Simple root constacyclic codes have been extensively study by many authors [3,4,6,7,9,13].…”

“…Using this results, A. Batoul et al considered the self-duality of cyclic codes over finite chain rings [2]. Some additionally necessary and sufficient conditions for the existence of non-trivial negacyclic and cyclic self-dual codes are given in [7] with a different method from that given in [4,9].…”

“…In this paper, we use idempotents of the quotient ring R[X]/ < X n − λ > to determine the structure of constacyclic codes over finite chain rings. Our method standardize the results of [2,4,7,9]. We first construct a complete set of primitive pairwise orthogonal idempotents of R[X]/ < g >, where R is a commutative finite local ring and g is a regular polynomial in R[X].…”