Two new families of two-weight codes

Abstract

We construct two new infinite families of trace codes of dimension $2m$, over the ring ${\mathbb{F}}_p+u{\mathbb{F}}_p,$ when $p$ is an odd prime. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. By Gray mapping, we obtain two infinite families of linear $p$-ary codes of respective lengths $(p^m-1{)}^2$ and $2(p^m-1{)}^2.$ When $m$ is singly-even, the first family gives five-weight codes. When $m$ is odd, and $p\equiv 3(\mathrm{mod}4),$ the first family yields $p$-ary two-weight codes, which are shown to be optimal by application of the Griesmer bound. The second family consists of two-weight codes that are shown to be optimal, by the Griesmer bound, whenever $p=3$ and $m\ge 3,$ or $p\ge 5$ and $m\ge 4.$ Applications to secret sharing schemes are given.

7 pages

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