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Several classes of $p$-ary linear codes with few-weights derived from Weil sums

Mrinal Kanti Bose, Abhay Kumar Singh

TL;DR

This work constructs new families of $p$-ary linear codes with few weights via a defining-set framework using two sets $D_u$ and $D^{'}$, enabling weight distributions to be explicitly determined through Weil sums and weakly regular bent function theory. From $D_u$ they obtain five classes of $4$-weight codes and one class of $2$-weight codes, while from $D^{'}$—where a weakly regular bent function with quadratic dual is employed—they derive $6$-weight, $8$-weight, and $9$-weight codes; all parameters (length, dimension, and weight distributions) are obtained by detailed sum evaluations. A notable outcome is an optimal two-weight code meeting the Griesmer bound for a special parameter choice, highlighting the practical relevance of these constructions. Overall, the paper broadens the catalog of explicit, non-equivalent few-weight codes over $\mathbb{F}_p$ by linking character sums to weight distributions, with potential applications in secret sharing, authentication codes, and combinatorial design theory.

Abstract

Linear codes with few weights have been a significant area of research in coding theory for many years, due to their applications in secret sharing schemes, authentication codes, association schemes, and strongly regular graphs. Inspired by the works of Cheng and Gao \cite{P8} and Wu, Li and Zeng \cite{P12}, in this paper, we propose several new classes of few-weight linear codes over the finite field $\mathbb{F}_{p}$ through the selection of two specific defining sets. Consequently, we obtain five classes of $4$-weight linear codes and one class of $2$-weight linear codes from our first defining set. Furthermore, by employing weakly regular bent functions in our second defining set, we derive two classes of $6$-weight codes, two classes of $8$-weight codes, and one class of $9$-weight codes. The parameters and weight distributions of all these constructed codes are wholly determined by detailed calculations on certain Weil sums over finite fields. In addition, we identify an optimal class of $2$-weight codes that meet the Griesmer bound.

Several classes of $p$-ary linear codes with few-weights derived from Weil sums

TL;DR

This work constructs new families of -ary linear codes with few weights via a defining-set framework using two sets and , enabling weight distributions to be explicitly determined through Weil sums and weakly regular bent function theory. From they obtain five classes of -weight codes and one class of -weight codes, while from —where a weakly regular bent function with quadratic dual is employed—they derive -weight, -weight, and -weight codes; all parameters (length, dimension, and weight distributions) are obtained by detailed sum evaluations. A notable outcome is an optimal two-weight code meeting the Griesmer bound for a special parameter choice, highlighting the practical relevance of these constructions. Overall, the paper broadens the catalog of explicit, non-equivalent few-weight codes over by linking character sums to weight distributions, with potential applications in secret sharing, authentication codes, and combinatorial design theory.

Abstract

Linear codes with few weights have been a significant area of research in coding theory for many years, due to their applications in secret sharing schemes, authentication codes, association schemes, and strongly regular graphs. Inspired by the works of Cheng and Gao \cite{P8} and Wu, Li and Zeng \cite{P12}, in this paper, we propose several new classes of few-weight linear codes over the finite field through the selection of two specific defining sets. Consequently, we obtain five classes of -weight linear codes and one class of -weight linear codes from our first defining set. Furthermore, by employing weakly regular bent functions in our second defining set, we derive two classes of -weight codes, two classes of -weight codes, and one class of -weight codes. The parameters and weight distributions of all these constructed codes are wholly determined by detailed calculations on certain Weil sums over finite fields. In addition, we identify an optimal class of -weight codes that meet the Griesmer bound.

Paper Structure

This paper contains 8 sections, 14 theorems, 48 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Exponential sums
  4. Cyclotomic fields $\mathbb{Q}(\zeta_{p})$
  5. Weakly regular bent functions
  6. Linear codes from the defining set $D_{u}$
  7. Linear codes from the defining set $D^{'}$
  8. Concluding remarks

Key Result

Lemma 1

PP15 Let $G(\eta_{m},\chi_{m})$ be defined as above, then where $p^{*}=\eta_{1}(-1)p=(-1)^{\frac{p-1}{2}}p$.

Theorems & Definitions (29)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Remark 1
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • proof
  • ...and 19 more