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：线性码的权分布与权层级是编码理论中的两个重要研究课题。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 $\mathbb{F}_{p^{s_1}}$ 上取值为 $\F_p$ 的二次形式（无论$f(x)$是否非退化），构造了一族三重量$p$元线性码，并确定了它们的权分布与权层级。其中大部分码可用于秘密共享方案。