Non-Invertible-Element Constacyclic Codes over Finite PIRs

TL;DR

This paper introduces and analyzes non-invertible-element constacyclic codes over finite principal ideal rings, providing algebraic structures, duality conditions, and constructing optimal codes with maximum minimum Hamming distances.

Contribution

It characterizes NIE-constacyclic codes over finite PIRs, including their algebraic structure, duals, and optimal code constructions, extending prior work to non-invertible elements.

Findings

Algebraic structure of NIE-constacyclic codes over finite chain rings

Necessary and sufficient conditions for duals to be NIE-constacyclic

Construction of optimal NIE-constacyclic codes achieving maximum minimum Hamming distance

Abstract

In this paper we introduce the notion of $\lambda$-constacyclic codes over finite rings $R$ for arbitary element $\lambda$ of $R$. We study the non-invertible-element constacyclic codes (NIE-constacyclic codes) over finite principal ideal rings (PIRs). We determine the algebraic structures of all NIE-constacyclic codes over finite chain rings, give the unique form of the sets of the defining polynomials and obtain their minimum Hamming distances. A general form of the duals of NIE-constacyclic codes over finite chain rings is also provided. In particular, we give a necessary and sufficient condition for the dual of an NIE-constacyclic code to be an NIE-constacyclic code. Using the Chinese Remainder Theorem, we study the NIE-constacyclic codes over finite PIRs. Furthermore, we construct some optimal NIE-constacyclic codes over finite PIRs in the sense that they achieve the maximum possible minimum Hamming distances for some given lengths and cardinalities.