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The weight distribution and weight hierarchy of a linear code are twoimportant 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-degenerateor not, we construct a family of three-weight $p$-ary linear codes anddetermine their weight distributions and weight hierarchies. Most of the codescan be used in secret sharing schemes.

Weight hierarchies of three-weight $p$-ary linear codes from inhomogeneous quadratic forms