Linear codes with eight weights over $\mathbb {F}_p+u\mathbb {F}_p$

Abstract

In this article, based on the defining set $D=\{x\in \mathbb {F}_{p^m}^*:Tr(x)=0\}$, we explore the Lee-weight distribution of linear codes ${\mathcal {C}}_D=\{(tr(ax^2))_{x\in D}:a\in \mathbb {F}_{p^m}+u\mathbb {F}_{p^m}\}$ over the finite ring $\mathbb {F}_p+u\mathbb {F}_p$ with p being an odd prime and $u^2=0$. By employing the exponential and Gauss sums, we calculate the Lee weight of all possible codewords as well as their frequencies. Two classes of eight-weight linear codes are obtained, where one of them is new. We also show that for some small values m, the code ${\mathcal {C}}_D$ has two weights ($m=2$) and seven weights ($m=3,4$ and $p=3$).