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We construct an infinite family of two-Lee-weight and three-Lee-weight codesover the chain ring $\mathbb{F}_p+u\mathbb{F}_p.$ They have the algebraicstructure of abelian codes. Their Lee weight distribution is computed by usingGauss sums. Then by using a linear Gray map, we obtain an infinite family ofabelian codes with few weights over $\mathbb{F}_p$. In particular, we obtain aninfinite family of two-weight codes which meets the Griesmer bound withequality. Finally, an application to secret sharing schemes is given.

Trace Codes with Few Weights over $\mathbb{F}_p+u\mathbb{F}_p$