Complete Weight Enumerators of a Family of Three-Weight Linear Codes

TL;DR

This paper derives the explicit complete weight enumerator for a family of three-weight p-ary linear codes, demonstrating their minimality and applicability in secret sharing schemes.

Contribution

It provides the first explicit complete weight enumerator for this family of codes, revealing their three-weight structure and minimal codeword property.

Findings

Codes are three-weight linear codes.

All nonzero codewords are minimal.

Codes are suitable for secret sharing.

Abstract

Linear codes have been an interesting topic in both theory and practice for many years. In this paper, for an odd prime $p$, we present the explicit complete weight enumerator of a family of $p$-ary linear codes constructed with defining set. The weight enumerator is an mmediate result of the complete weight enumerator, which shows that the codes proposed in this paper are three-weight linear codes. Additionally, all nonzero codewords are minimal and thus they are suitable for secret sharing.