Linear codes with few weights over $\mathbb{F}_2+u\mathbb{F}_2$

TL;DR

This paper constructs an infinite family of five-weight codes over a specific ring, analyzes their weight distribution using character sums, and explores applications in secret sharing schemes.

Contribution

It introduces a new class of five-weight codes over the ring F_2+uF_2 with detailed weight distribution and support structure analysis.

Findings

Weight distribution of Gray images determined

Support structure characterized

Application to secret sharing schemes provided

Abstract

In this paper, we construct an infinite family of five-weight codes from trace codes over the ring $R=\mathbb{F}_2+u\mathbb{F}_2$, where $u^2=0.$ The trace codes have the algebraic structure of abelian codes. Their Lee weight is computed by using character sums. Combined with Pless power moments and Newton's Identities, the weight distribution of the Gray image of trace codes was present. Their support structure is determined. An application to secret sharing schemes is given.