Optimal three-weight cubic codes

Abstract

In this paper, we construct an infinite family of three-weight binary codes from linear codes over the ring $R={\mathbb{F}}_2+v{\mathbb{F}}_2+v^2{\mathbb{F}}_2$, where $v^3=1.$ These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distributions are computed by employing character sums. The three-weight binary linear codes which we construct are shown to be optimal when $m$ is odd and $m>1$. They are cubic, that is to say quasi-cyclic of co-index three. An application to secret sharing schemes is given.

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