Primitive Idempotents and Constacyclic Codes over Finite Chain Rings

TL;DR

This paper develops a detailed algebraic framework for decomposing rings over finite chain rings using primitive idempotents, enabling a comprehensive understanding of the structure, duals, and self-duality of constacyclic codes.

Contribution

It constructs primitive idempotents in polynomial quotient rings over finite chain rings and applies this to analyze the structure and duality of constacyclic codes.

Findings

Explicit construction of primitive idempotents in $R[X]/<g>$

Decomposition of constacyclic codes over finite chain rings

Characterization of self-dual constacyclic codes

Abstract

Let $R$ be a commutative local finite ring. In this paper, we construct the complete set of pairwise orthogonal primitive idempotents of $R[X]/<g>$ where $g$ is a regular polynomial in $R[X]$. We use this set to decompose the ring $R[X]/<g>$ and to give the structure of constacyclic codes over finite chain rings. This allows us to describe generators of the dual code $\mathcal{C}^\bot$ of a constacyclic code $\mathcal{C}$ and to characterize non-trivial self-dual constacyclic codes over finite chain rings.