Generalized Hamming Weights of Linear Codes from Quadratic Forms over Finite Fields of Even Characteristic

TL;DR

This paper determines the complete generalized Hamming weights of linear codes derived from quadratic forms over finite fields of even characteristic, extending previous results from odd characteristic fields and deepening understanding of code structure.

Contribution

It provides a new analysis of quadratic forms over even characteristic fields to find the weight hierarchy of associated linear codes, complementing prior work on odd characteristic fields.

Findings

Complete weight hierarchy for codes from quadratic forms over even characteristic fields.

Analysis of quadratic form behavior on subspaces and dual spaces.

Extension of previous odd characteristic results to even characteristic case.

Abstract

The generalized Hamming weight of linear codes is a natural generalization of the minimum Hamming distance. They convey the structural information of a linear code and determine its performance in various applications, and have become one of important research topics in coding theory. Recently, Li (IEEE Trans. Inf. Theory, 67(1): 124-129, 2021) and Li and Li (Discrete Math., 345: 112718, 2022) obtained the complete weight hierarchy of linear codes from a quadratic form over a finite field of odd characteristic by analysis of the solutions of the restricted quadratic equation in its subspace. In this paper, we further determine the complete weight hierarchy of linear codes from a quadratic form over a finite field of even characteristic by carefully studying the behavior of the quadratic form on the subspaces of this field and its dual space, and complement the results of Li and Li.