Complete weight enumerators and weight hierarchies for linear codes from quadratic forms
Xiumei Li, Xiaotong Sun, Min Sha
TL;DR
This work constructs two families of linear codes from quadratic forms using a bivariate framework and determines their complete weight enumerators and weight hierarchies exactly via exponential sums. It extends prior constructions by providing explicit CWE formulas, GHWs, and Griesmer-optimality conditions, along with minimality results. The descent procedure to prime fields yields additional p-ary codes with computable weight structures, preserving optimality in many cases. The results enhance understanding of codes from quadratic forms and offer potential applications in authentication, secret sharing, and related cryptographic design spaces.
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 $\mathcal{C}_{Q}'$.