The Lee weight distributions of several classes of linear codes over $\mathbb{Z}_4$

TL;DR

This paper constructs new linear codes over the ring of integers modulo 4 using Galois rings and trace functions, determining their Lee weight distributions and identifying codes with optimal minimum Lee distances.

Contribution

It introduces a defining-set approach leveraging Galois ring properties to construct and analyze new $ ext{Z}_4$-linear codes with good parameters.

Findings

Derived new $ ext{Z}_4$-linear codes with optimal Lee distances

Determined Lee weight distributions for several code classes

Identified codes with best known minimum Lee distances

Abstract

Let $\mathbb{Z}_4$ denote the ring of integers modulo $4$. The Galois ring GR$(4,m)$, which consists of $4^m$ elements, represents the Galois extension of degree $m$ over $\mathbb{Z}_4$. The constructions of codes over $\mathbb{Z}_4$ have garnered significant interest in recent years. In this paper, building upon previous research, we utilize the defining-set approach to construct several classes of linear codes over $\mathbb{Z}_4$ by effectively using the properties of the trace function from GR$(4,m)$ to $\mathbb{Z}_4$. As a result, we have been able to obtain new linear codes over $\mathbb{Z}_4$ with good parameters and determine their Lee weight distributions. Upon comparison with the existing database of $\mathbb{Z}_4$ codes, our construction can yield novel linear codes, as well as linear codes that possess the best known minimum Lee distance.