For any positive integer $m$ and an odd prime $p$; let $\mathbb{F}_{q}+u\mathbb{F}_{q}$, where $q=p^{m}$, be a ring extension of the ring $\mathbb{F}_{p}+u\mathbb{F}_{p}.$ In this paper, we construct linear codes over $\mathbb{F}_{p}+u\mathbb{F}_{p}$ by using trace function defined on $\mathbb{F}_{q}+u\mathbb{F}_{q}$ and determine their Hamming weight distributions by employing symplectic-weight distributions of their Gray images.

翻译：对于任意正整数$m$和奇素数$p$，令$\mathbb{F}_{q}+u\mathbb{F}_{q}$（其中$q=p^{m}$）为环$\mathbb{F}_{p}+u\mathbb{F}_{p}$的环扩张。本文利用定义在$\mathbb{F}_{q}+u\mathbb{F}_{q}$上的迹函数，构造了$\mathbb{F}_{p}+u\mathbb{F}_{p}$上的线性码，并通过其Gray像的辛重量分布确定了这些码的汉明重量分布。