Three-weight codes and the quintic construction

Abstract

We construct a class of three-Lee-weight and two infinite families of five-Lee-weight codes over the ring $R={\mathbb{F}}_2+v{\mathbb{F}}_2+v^2{\mathbb{F}}_2+v^3{\mathbb{F}}_2+v^4{\mathbb{F}}_2,$ where $v^5=1.$ The same ring occurs in the quintic construction of binary quasi-cyclic codes. %The length of these codes depends on the degree $m$ of ring extension. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Given a linear Gray map, we obtain three families of binary abelian codes with few weights. In particular, we obtain a class of three-weight codes which are optimal. Finally, an application to secret sharing schemes is given.

15 pages, submitted on 21 November, 2016. arXiv admin note: text overlap with arXiv:1612.00118