Repeated-root constacyclic codes over the finite chain ring $\mathbf{ \mathbb{F}_{p^m}[u]/\langle u^3 \rangle }$

TL;DR

This paper classifies all repeated-root constacyclic codes over a specific finite chain ring, detailing their sizes, duals, distances, and weight distributions, and provides examples of isodual codes.

Contribution

It fully characterizes repeated-root constacyclic codes over the ring _{p^m}[u]/rac{u^3}{} with explicit parameters and properties, including duals and weight distributions.

Findings

All such codes are classified and their sizes determined.

Explicit dual codes and isodual codes are listed.

Hamming and RT distances, along with weight distributions, are computed.

Abstract

Let $\mathcal{R}=\mathbb{F}_{p^m}[u]/\langle u^3 \rangle $ be the finite commutative chain ring with unity, where $p$ is a prime, $m$ is a positive integer and $\mathbb{F}_{p^m}$ is the finite field with $p^m$ elements. In this paper, we determine all repeated-root constacyclic codes of arbitrary lengths over $\mathcal{R},$ their sizes and their dual codes. As an application, we list some isodual constacyclic codes over $\mathcal{R}.$ We also determine Hamming distances, RT distances, and RT weight distributions of some repeated-root constacyclic codes over $\mathcal{R}.$