Two new families of two-weight codes

TL;DR

This paper introduces two new infinite families of two-weight codes over a ring, analyzes their weight distributions, and demonstrates their optimality and applications in secret sharing schemes.

Contribution

The paper constructs and characterizes two new families of trace codes over a ring, including their weight distributions and optimality conditions, with applications to secret sharing.

Findings

First family yields five-weight codes when m is singly-even.

First family produces two-weight codes for p ≡ 3 mod 4 when m is odd.

Second family consists of optimal two-weight codes under certain conditions.

Abstract

We construct two new infinite families of trace codes of dimension $2m$, over the ring $\mathbb{F}_p+u\mathbb{F}_p,$ when $p$ is an odd prime. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. By Gray mapping, we obtain two infinite families of linear $p$-ary codes of respective lengths $(p^m-1)^2$ and $2(p^m-1)^2.$ When $m$ is singly-even, the first family gives five-weight codes. When $m$ is odd, and $p\equiv 3 \pmod{4},$ the first family yields $p$-ary two-weight codes, which are shown to be optimal by application of the Griesmer bound. The second family consists of two-weight codes that are shown to be optimal, by the Griesmer bound, whenever $p=3$ and $m \ge 3,$ or $p\ge 5$ and $m\ge 4.$ Applications to secret sharing schemes are given.