  In this paper, we construct an infinite family of five-weight codes from
trace codes over the ring $R=\mathbb{F}_2+u\mathbb{F}_2$, where $u^2=0.$ The
trace codes have the algebraic structure of abelian codes. Their Lee weight is
computed by using character sums. Combined with Pless power moments and
Newton's Identities, the weight distribution of the Gray image of trace codes
was present. Their support structure is determined. An application to secret
sharing schemes is given.
