Three-weight codes and the quintic construction

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.