New few weight codes from trace codes over a local Ring

In this paper, new few weights linear codes over the local ring $R=\mathbb{F}_p+u\mathbb{F}_p+v\mathbb{F}_p+uv\mathbb{F}_p,$ with $u^2=v^2=0, uv=vu,$ are constructed by using the trace function defined over an extension ring of degree $m.$ %In fact, These codes are punctured from the linear code is defined in \cite{SWLP} up to coordinate permutations. These trace codes have the algebraic structure of abelian codes. Their weight distributions are evaluated explicitly by means of Gaussian sums over finite fields. Two different defining sets are explored. Using a linear Gray map from $R$ to $\mathbb{F}_p^4,$ we obtain several families of new $p$-ary codes from trace codes of dimension $4m$. For the first defining set: when $m$ is even, or $m$ is odd and $p\equiv3 ~({\rm mod} ~4),$ we obtain a new family of two-weight codes, which are shown to be optimal by the application of the Griesmer bound; when $m$ is even and under some special conditions, we obtain two new classes of three-weight codes. For the second defining set: we obtain a new class of two-weight codes and prove that it meets the Griesmer bound. In addition, we give the minimum distance of the dual code. Finally, applications of the $p$-ary image codes in secret sharing schemes are presented.