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Some three-weight linear codes and their complete weight enumerators and weight hierarchies

Xiumei Li, Zongxi Chen, Fei Li

TL;DR

This paper constructs a class of three-weight Fq from quadratic functions via a bivariate construction and determines the complete weight enumerators and weight hierarchies of these linear codes completely.

Abstract

Linear codes with a few weights can be applied to secrete sharing, authentication codes, association schemes and strongly regular graphs. For an odd prime power $q$, we construct a class of three-weight $\F_q$-linear codes from quadratic functions via a bivariate construction and then determine the complete weight enumerators and weight hierarchies of these linear codes completely. This paper generalizes some results in Li et al. (2022) and Hu et al. (2024).

Some three-weight linear codes and their complete weight enumerators and weight hierarchies

TL;DR

This paper constructs a class of three-weight Fq from quadratic functions via a bivariate construction and determines the complete weight enumerators and weight hierarchies of these linear codes completely.

Abstract

Linear codes with a few weights can be applied to secrete sharing, authentication codes, association schemes and strongly regular graphs. For an odd prime power , we construct a class of three-weight -linear codes from quadratic functions via a bivariate construction and then determine the complete weight enumerators and weight hierarchies of these linear codes completely. This paper generalizes some results in Li et al. (2022) and Hu et al. (2024).
Paper Structure (7 sections, 16 theorems, 74 equations, 3 tables)

This paper contains 7 sections, 16 theorems, 74 equations, 3 tables.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Some notations fixed throughout this paper
  4. Quadratic form
  5. The complete weight enumerators
  6. The weight hierarchies of the presented linear codes
  7. Concluding Remarks

Key Result

Lemma 2.1

With the symbols and notations above and let $f(x)=a_2x^2+a_1x+a_0\in\mathbb{F}_q[x]$, where $a_2\neq 0,q=p^m$. Then (1) $\sum\limits_{c\in\mathbb{F}_q^*}\eta(c)\zeta_p^{\mathrm{Tr}_p^q(c)}=(-1)^{m-1}\eta(-1)p^m(p^*)^{-\frac{m}{2}}$. (2) $\sum\limits_{c\in\mathbb{F}_q}\zeta_p^{\mathrm{Tr}_p^q(f(c))}

Theorems & Definitions (31)

  • Lemma 2.1: LN97
  • Lemma 2.2
  • Lemma 2.3: LF21
  • Lemma 2.4: TXF17
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 21 more