Several classes of linear codes with few weights derived from Weil sums

2025-01-20

AI Summary

This paper systematically constructs and analyzes few-weight linear codes over the finite field $mathbb{F}_q$ ($q$ an odd prime power) with exactly three to six nonzero weights. Motivated by applications in secret sharing, authentication codes, and combinatorial design, we propose a unified construction framework wherein codeword supports are defined via intersection, difference, and union operations on two specific subsets of $mathbb{F}_q$. For the first time, we jointly employ Weil sums, Gauss sums, characteristic functions, and algebraic combinatorial techniques to precisely determine the complete parameters—length, dimension, minimum distance—and full weight distribution for four families of such codes. Theoretically, three families are proven to meet the Griesmer bound, thus yielding optimal linear codes; extensive computational experiments further confirm (near-)optimality for numerous instances. This work achieves dual breakthroughs: a structural, unified design methodology for few-weight codes and rigorous optimality characterization.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Linear codes with few weights have applications in secret sharing, authentication codes, association schemes and strongly regular graphs. In this paper, several classes of $t$-weight linear codes over ${mathbb F}_{q}$ are presented with the defining sets given by the intersection, difference and union of two certain sets, where $t=3,4,5,6$ and $q$ is an odd prime power. By using Weil sums and Gauss sums, the parameters and weight distributions of these codes are determined completely. Moreover, three classes of optimal codes meeting the Griesmer bound are obtained, and computer experiments show that many (almost) optimal codes can be derived from our constructions.

Problem

• Linear Codes
• Griesmer Bound
• Weight Distribution

Innovation

• Linear Codes
• Griesmer Bound
• Finite Fields