The weight distribution and weight hierarchy of a linear code are two important research topics in coding theory. In this paper, choosing $ D=\Big\{(x,y)\in \Big(\F_{p^{s_1}}\times\F_{p^{s_2}}\Big)\Big\backslash\{(0,0)\}: f(x)+\Tr_1^{s_2}(\alpha y)=0\Big\}$ as a defining set , where $\alpha\in\mathbb{F}_{p^{s_2}}^*$ and $f(x)$ is a quadratic form over $\mathbb{F}_{p^{s_1}}$ with values in $\F_p$, whether $f(x)$ is non-degenerate or not, we construct a family of three-weight $p$-ary linear codes and determine their weight distributions and weight hierarchies. Most of the codes can be used in secret sharing schemes.

翻译：线性码的重量分布和重量谱系是编码理论中两个重要的研究课题。本文选取$ D=\Big\{(x,y)\in \Big(\F_{p^{s_1}}\times\F_{p^{s_2}}\Big)\Big\backslash\{(0,0)\}: f(x)+\Tr_1^{s_2}(\alpha y)=0\Big\}$作为定义集，其中$\alpha\in\mathbb{F}_{p^{s_2}}^*$且$f(x)$是$\mathbb{F}_{p^{s_1}}$上取值于$\F_p$的二次型，无论$f(x)$是否非退化，我们构造了一族三重量$p$元线性码，并确定了其重量分布和重量谱系。这些码中的大部分可用于秘密共享方案。