Constructions For Several Few-weight Linear Codes And Their Applications

TL;DR

This paper constructs new classes of few-weight linear codes over finite fields, explicitly determines their weight enumerators, and explores their applications in secret sharing and combinatorial designs.

Contribution

It introduces several new classes of linear codes with specific weight distributions for odd primes and integers, including MDS codes, and analyzes their applications.

Findings

Explicit formulas for weight enumerators using Gauss sums

Construction of MDS codes with parameters [p,3,p-2]

Codes applicable in secret sharing and s-sum sets

Abstract

In this paper, for any odd prime $p$ and an integer $m\ge 3$, several classes of linear codes with $t$-weight $(t=3,5,7)$ are obtained based on some defining sets, and then their complete weight enumerators are determined explicitly by employing Gauss sums and quadratic character sums. Especially for $m = 3$, a class of MDS codes with parameters $[p,3,p-2]$ are obtained. Furthermore, some of these codes can be suitable for applications in secret sharing schemes and $s$-sum sets for any odd $s>1$.