Constacyclic Codes over Commutative Finite Principal Ideal Rings

TL;DR

This paper studies constacyclic codes over finite commutative principal ideal rings, providing explicit generator polynomials, BCH bounds, and conditions for codes to be principal, advancing algebraic coding theory over rings.

Contribution

It explicitly constructs generator and check polynomials for constacyclic codes over such rings and establishes conditions for their principality, including a BCH bound.

Findings

Constacyclic codes are principal under certain conditions.

Explicit generator and check polynomials are constructed.

A BCH bound for these codes is established.

Abstract

For any constacyclic code over a finite commutative chain ring of length coprime to the characteristic of the ring, we construct explicitly generator polynomials and check polynomials, and exhibit a BCH bound for such constacyclic codes. As a consequence, such constacyclic codes are principal. Further, we get a necessary and sufficient condition that the cyclic codes over a finite commutative principal ideal ring are all principal. This condition is still sufficient for constacyclic codes over such rings being principal.