Weight hierarchies of a family of linear codes associated with degenerate quadratic forms

TL;DR

This paper investigates the weight hierarchies of linear codes derived from degenerate quadratic forms over finite fields by analyzing subspaces and quotient spaces to understand their structure.

Contribution

It introduces a method to determine the weight hierarchies of linear codes associated with degenerate quadratic forms using quotient space analysis.

Findings

Derived explicit formulas for weight hierarchies.

Connected quadratic form degeneracy to code properties.

Provided new insights into code structure related to quadratic forms.

Abstract

We restrict a degenerate quadratic form $f$ over a finite field of odd characteristic to subspaces. Thus, a quotient space related to $f$ is introduced. Then we get a non-degenerate quadratic form induced by $f$ over the quotient space. Some related results on the subspaces and quotient space are obtained. Based on this, we solve the weight hierarchies of a family of linear codes related to $f.$