Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Zhexin Wang, Nian Li, Xiangyong Zeng, Xiaohu Tang

TL;DR

This work investigates the Lee weight distributions of several classes of linear codes over $\mathbb{Z}_4$ constructed via a defining-set approach on the Galois ring $GR(4,m)$. Using the Teichmüller set $\mathbb{L}$ and the trace $\mathrm{Tr}_1^m$, it derives explicit Lee weight distributions for odd $m$ by analyzing two exponential sums $S_+(u)$ and $S_-(u)$ and their interactions. The authors identify numerous $\mathbb{Z}_4$-linear codes with good or best-known minimum Lee distance and show many are new relative to existing databases. For even $m$, weight distributions remain open, defining a clear direction for further research with potential impact on the design of efficient error-correcting codes over $\mathbb{Z}_4$.

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.

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

TL;DR

This work investigates the Lee weight distributions of several classes of linear codes over constructed via a defining-set approach on the Galois ring . Using the Teichmüller set and the trace , it derives explicit Lee weight distributions for odd by analyzing two exponential sums and and their interactions. The authors identify numerous -linear codes with good or best-known minimum Lee distance and show many are new relative to existing databases. For even , weight distributions remain open, defining a clear direction for further research with potential impact on the design of efficient error-correcting codes over .

Abstract

Let denote the ring of integers modulo . The Galois ring GR, which consists of elements, represents the Galois extension of degree over . The constructions of codes over 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 by effectively using the properties of the trace function from GR to . As a result, we have been able to obtain new linear codes over with good parameters and determine their Lee weight distributions. Upon comparison with the existing database of codes, our construction can yield novel linear codes, as well as linear codes that possess the best known minimum Lee distance.

Paper Structure

This paper contains 9 sections, 13 theorems, 70 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Auxiliary lemmas
  4. Proof of Theorem \ref{['thm1']}
  5. Proof of Theorem \ref{['thm2']}
  6. Proof of Theorem \ref{['thm3']}
  7. Conclusion
  8. Proof of the Lemma \ref{['lem-S123']}
  9. Proof of the Lemma \ref{['lem-S1234']}

Key Result

Theorem 1

Let $m>3$ be odd, $\mathcal{C}_D$ and $D_t$ defined as eq-code and eq-Dt respectively, and $D=D_t$. Then for $t=0,1,2,3$, $\mathcal{C}_D$ has parameters as follows: Moreover, the Lee weight distribution of $\mathcal{C}_D$ for $t=0$ is given by the Lee weight distribution of $\mathcal{C}_D$ for $t=2$ is given by and the Lee weight distributions of $\mathcal{C}_D$ for $t=1,3$ are given by

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Remark 2
  • Remark 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 13 more