The weight hierarchies of three classes of linear codes

TL;DR

This paper determines the complete weight hierarchies of three classes of linear codes over finite fields by analyzing their defining sets and intersections with dual subspaces, advancing understanding of their structural properties.

Contribution

It introduces a method to explicitly compute the generalized Hamming weights for these codes, which was previously challenging.

Findings

Complete weight hierarchies are determined for the three code classes.

The method involves analyzing intersections with dual subspaces.

Results provide new insights into the structure of these codes.

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.