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Several classes of three-weight or four-weight linear codes

Qunying Liao, Zhaohui Zhang, Peipei Zheng

TL;DR

This work develops a defining-set framework to construct few-weight projective binary linear codes over $\\mathbb{F}_{2}$ using trace forms. By employing additive character sums, it derives exact weight distributions for three code families $C_D1$, $C_D2$, and $C_D3$ defined by sets in $\\mathbb{F}_{2^m}^* \\times \\mathbb{F}_{2^m}$ with trace constraints, and proves projectivity and minimality where applicable. The main results include a four-weight projective code from $C_{D1}$ with explicit parameters and weights, a three-weight code from $C_{D2}$ with its parameters, and a second four-weight projective code from $C_{D3}$, all validated through detailed exponential-sum evaluations. These codes have potential applications in secret sharing schemes (notably the first two), and in at least one case (the $m=3$ instance) the Griesmer bound confirms optimality. The work advances the catalog of explicit constructions of binary, projective, few-weight codes with rigorous weight distributions.

Abstract

In this manuscript, we construct a class of projective three-weight linear codes and two classes of projective four-weight linear codes over F2 from the defining sets construction, and determine their weight distributions by using additive characters. Especially, the projective three-weight linear code and one class of projective four-weight linear codes (Theorem 4.1) can be applied in secret sharing schemes.

Several classes of three-weight or four-weight linear codes

TL;DR

This work develops a defining-set framework to construct few-weight projective binary linear codes over using trace forms. By employing additive character sums, it derives exact weight distributions for three code families , , and defined by sets in with trace constraints, and proves projectivity and minimality where applicable. The main results include a four-weight projective code from with explicit parameters and weights, a three-weight code from with its parameters, and a second four-weight projective code from , all validated through detailed exponential-sum evaluations. These codes have potential applications in secret sharing schemes (notably the first two), and in at least one case (the instance) the Griesmer bound confirms optimality. The work advances the catalog of explicit constructions of binary, projective, few-weight codes with rigorous weight distributions.

Abstract

In this manuscript, we construct a class of projective three-weight linear codes and two classes of projective four-weight linear codes over F2 from the defining sets construction, and determine their weight distributions by using additive characters. Especially, the projective three-weight linear code and one class of projective four-weight linear codes (Theorem 4.1) can be applied in secret sharing schemes.

Paper Structure

This paper contains 8 sections, 14 theorems, 71 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some necessary Lemmas
  4. Our main results and their proofs
  5. Our main results
  6. The proofs of Theorems 4.1-4.3
  7. Some examples
  8. Conclusions

Key Result

Lemma 2.1

A47 (a) For $\alpha \in \mathbb{F}_{2^{m}}$, the trace function $\operatorname{Tr}(\alpha)$ of $\alpha$ over $\mathbb{F}_{2^{m}}$ is defined by (b) when $m$ is odd, $\operatorname{Tr}(1)=1;$ (c) $\operatorname{Tr}(\alpha+\beta)=\operatorname{Tr}(\alpha)+\operatorname{Tr}(\beta)$ for all $\alpha, \beta \in \mathbb{F}_{2^{m}}.$

Theorems & Definitions (22)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 12 more