Two-weight and three-weight codes from trace codes over \mathbbF_p+u\mathbbF_p+v\mathbbF_p+uv\mathbbF_p

2016/12/01 by Yan Liu, Liu, Yan, Minjia Shi +3 · 1 citation
Computer Science · Engineering · #Coding theory and cryptography #Cooperative Communication and Network Coding #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1612.00118

Abstract

We construct an infinite family of two-Lee-weight and three-Lee-weight codes over the non-chain ring \mathbbF_p+u\mathbbF_p+v\mathbbF_p+uv\mathbbF_p, where u²=0,v²=0,uv=vu. These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. With a linear Gray map, we obtain a class of abelian three-weight codes and two-weight codes over \mathbbF_p. In particular, the two-weight codes we describe are shown to be optimal by application of the Griesmer bound. We also discuss their dual Lee distance. Finally, an application to secret sharing schemes is given.

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