Complete weight enumerators and weight hierarchies for linear codes from quadratic forms

TL;DR

This paper constructs new linear codes from quadratic forms over finite fields, determines their complete weight enumerators and hierarchies, and identifies some as minimal or optimal, advancing coding theory knowledge.

Contribution

It introduces two classes of linear codes from quadratic forms, fully determines their weight properties, and explores their minimality and optimality.

Findings

Codes have at most four nonzero weights

Most codes are minimal, some meet Griesmer bound

Weight hierarchies are explicitly determined

Abstract

In this paper, for an odd prime power $q$, we extend the construction of Xie et al. \cite{XOYM2023} to propose two classes of linear codes $\mathcal{C}_{Q}$ and $\mathcal{C}_{Q}'$ over the finite field $\mathbb{F}_{q}$ with at most four nonzero weights. These codes are derived from quadratic forms through a bivariate construction. We completely determine their complete weight enumerators and weight hierarchies by employing exponential sums. Most of these codes are minimal and some are optimal in the sense that they meet the Griesmer bound. Furthermore, we also establish the weight hierarchies of $\mathcal{C}_{Q,N}$ and $\mathcal{C}_{Q,N}'$, which are the descended codes of $\mathcal{C}_{Q}$ and $\mathcal{C}_{Q}'$.