A Note on Hamming distance of constacyclic codes of length $p^s$ over $\mathbb F_{p^m} + u\mathbb F_{p^m}$

TL;DR

This paper determines the minimum Hamming distances of cyclic codes of length p^s over a specific ring and establishes an isometry linking cyclic and constacyclic codes, enhancing understanding of their structure.

Contribution

It provides explicit minimum Hamming distances for all cyclic codes over the ring and introduces an isometry connecting cyclic and constacyclic codes of length p^s.

Findings

Minimum Hamming distances of cyclic codes over ${ m R}$ are fully characterized.

An isometry between cyclic and $eta$-constacyclic codes is established.

Results facilitate code classification and analysis over the ring.

Abstract

For any prime $p$, $\lambda$-constacyclic codes of length $p^s$ over ${\cal R}=\mathbb{F}_{p^m} + u\mathbb{F}_{p^m}$ are precisely the ideals of the local ring ${\cal R}_{\lambda}=\frac{{\cal R}[x]}{\left\langle x^{p^s}-\lambda \right\rangle}$, where $u^2=0$. In this paper, we first investigate the Hamming distances of cyclic codes of length $p^s$ over ${\cal R}$. The minimum Hamming distances of all cyclic codes of length $p^s$ over ${\cal R}$ are determined. Moreover, an isometry between cyclic and $\alpha$-constacyclic codes of length $p^s$ over ${\cal R}$ is established, where $\alpha$ is a nonzero element of $\mathbb{F}_{p^m}$, which carries over the results regarding cyclic codes corresponding to $\alpha$-constacyclic codes of length $p^s$ over ${\cal R}$.