Some ternary cubic two-weight codes

TL;DR

This paper constructs and analyzes new classes of optimal two-weight ternary codes derived from trace codes over a specific algebraic structure, with applications in secret sharing.

Contribution

It introduces two new infinite families of optimal two-weight ternary codes from trace codes over a subgroup of a ring, expanding coding theory and cryptography applications.

Findings

Codes have three nonzero weights when m is singly-even.

Two new infinite families of two-weight codes are constructed for odd m.

Codes are optimal and applicable to secret sharing schemes.

Abstract

We study trace codes with defining set $L,$ a subgroup of the multiplicative group of an extension of degree $m$ of the alphabet ring $\mathbb{F}_3+u\mathbb{F}_3+u^{2}\mathbb{F}_{3},$ with $u^{3}=1.$ These codes are abelian, and their ternary images are quasi-cyclic of co-index three (a.k.a. cubic codes). Their Lee weight distributions are computed by using Gauss sums. These codes have three nonzero weights when $m$ is singly-even and $|L|=\frac{3^{3m}-3^{2m}}{2}.$ When $m$ is odd, and $|L|=\frac{3^{3m}-3^{2m}}{2}$, or $|L|={3^{3m}-3^{2m}}$ and $m$ is a positive integer, we obtain two new infinite families of two-weight codes which are optimal. Applications of the image codes to secret sharing schemes are also given.