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.
Generalized Rank Weight and Extended Generalized Poset Weight Defined For Codes Over Rings: A Galois Connection Approach
Fudan University（复旦大学）；National and Local Joint Laboratory of Cyberspace Security Technology（网络空间安全技术国家地方联合工程实验室）；Yiwu Research Institute of Fudan University（复旦大学义乌研究院）；The University of Hong Kong（香港大学）
