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 = F2 + vF2 + v 2 F2 + v 3 F2 + v 4 F2, where v 5 = 1. The same ring occurs in the quintic construction of binary quasi-cyclic codes. 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.