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The weight hierarchies of three classes of linear codes

Wei Lu, Qingyao Wang, Xiaoqiang Wang, Dabin Zheng

TL;DR

The paper addresses the challenge of determining complete weight hierarchies for three classes of linear codes constructed from defining sets over finite fields. It develops a unified approach based on analyzing the intersections of defining sets with dual subspaces and constructing r-dimensional subspaces that attain derived bounds, with analysis supported by exponential-sum techniques. Explicit closed-form expressions for the r-th generalized Hamming weight are obtained for each class: unitary-form, bivariate-form, and butterfly-structure codes, under appropriate parameter conditions. The results fill gaps in the literature on weight hierarchies for defining-set codes and provide precise tools for assessing code performance in applications sensitive to higher-weight structure.

Abstract

Studying the generalized Hamming weights of linear codes is a significant research area within coding theory, as it provides valuable structural information about the codes and plays a crucial role in determining their performance in various applications. However, determining the generalized Hamming weights of linear codes, particularly their weight hierarchy, is generally a challenging task. In this paper, we focus on investigating the generalized Hamming weights of three classes of linear codes over finite fields. These codes are constructed by different defining sets. By analysing the intersections between the definition sets and the duals of all $r$-dimensional subspaces, we get the inequalities on the sizes of these intersections. Then constructing subspaces that reach the upper bounds of these inequalities, we successfully determine the complete weight hierarchies of these codes.

The weight hierarchies of three classes of linear codes

TL;DR

The paper addresses the challenge of determining complete weight hierarchies for three classes of linear codes constructed from defining sets over finite fields. It develops a unified approach based on analyzing the intersections of defining sets with dual subspaces and constructing r-dimensional subspaces that attain derived bounds, with analysis supported by exponential-sum techniques. Explicit closed-form expressions for the r-th generalized Hamming weight are obtained for each class: unitary-form, bivariate-form, and butterfly-structure codes, under appropriate parameter conditions. The results fill gaps in the literature on weight hierarchies for defining-set codes and provide precise tools for assessing code performance in applications sensitive to higher-weight structure.

Abstract

Studying the generalized Hamming weights of linear codes is a significant research area within coding theory, as it provides valuable structural information about the codes and plays a crucial role in determining their performance in various applications. However, determining the generalized Hamming weights of linear codes, particularly their weight hierarchy, is generally a challenging task. In this paper, we focus on investigating the generalized Hamming weights of three classes of linear codes over finite fields. These codes are constructed by different defining sets. By analysing the intersections between the definition sets and the duals of all -dimensional subspaces, we get the inequalities on the sizes of these intersections. Then constructing subspaces that reach the upper bounds of these inequalities, we successfully determine the complete weight hierarchies of these codes.
Paper Structure (9 sections, 12 theorems, 98 equations)

This paper contains 9 sections, 12 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear codes with the unitary form
  4. Linear codes with the bivariate form
  5. The weight hierarchies of three classes of linear Codes
  6. Linear codes from the defining set ${\mathbb F}_{q^m}\setminus\bigcup_{i=0}^{h} (\theta_i+{\mathbb F}_{q^{k}})$
  7. Linear codes from the defining-set $({\mathbb F}_{q^m}\setminus {\mathbb F}_{q^s})\times({\mathbb F}_{q^k}\setminus {\mathbb F}_{q^\ell})$
  8. Linear codes from a defining-set of butterfly structure
  9. Concluding remarks

Key Result

Proposition 1

Likangquan2022 Let $K$ be as mentioned in (eq:0522), and let ${\mathcal{C}}_{D}$ be a linear code defined in (linearcode1) with dimension $m-{\rm dim}_{{\mathbb F}_q} K$. Then, for any integer $1\leq r\leq m-{\rm dim}_{{\mathbb F}_q} K$, the $r$-th generalized Hamming weight of ${\mathcal{C}}_{D}$ i where $[{\mathbb F}_{q^m}, r]$ denotes the set of all $r$-dimensional subspaces of ${\mathbb F}_{q^

Theorems & Definitions (30)

  • Proposition 1
  • Lemma 2
  • Lemma 3
  • Proposition 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • Remark 7
  • ...and 20 more