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arXiv · 1612.00126

Three-weight codes and the quintic construction

Yan Liu·Minjia Shi·Patrick Solé
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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.

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Yan Liu, Minjia Shi, Patrick Solé. 2016-12-01. Three-weight codes and the quintic construction. https://arxiv.org/abs/1612.00126

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