Trace Codes with Few Weights over $\mathbb{F}_p+u\mathbb{F}_p$

TL;DR

This paper constructs infinite families of two- and three-weight abelian codes over a chain ring, computes their weight distributions using Gauss sums, and applies them to secret sharing schemes.

Contribution

It introduces new algebraic codes over a chain ring with few weights and demonstrates their optimality and application in secret sharing.

Findings

Constructed infinite families of two- and three-weight codes.

Computed weight distributions using Gauss sums.

Achieved codes meeting the Griesmer bound with equality.

Abstract

We construct an infinite family of two-Lee-weight and three-Lee-weight codes over the chain ring $\mathbb{F}_p+u\mathbb{F}_p.$ They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. Then by using a linear Gray map, we obtain an infinite family of abelian codes with few weights over $\mathbb{F}_p$. In particular, we obtain an infinite family of two-weight codes which meets the Griesmer bound with equality. Finally, an application to secret sharing schemes is given.