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

The weight distribution and weight hierarchy of a linear code are two important research topics in coding theory. In this paper, choosing $D=\{(x,y)\in ({\text{F}}_{p^{s_1}}\times {\text{F}}_{p^{s_2}})\backslash \{(0,0)\}:f(x)+{\text{Tr}}_1^{s_2}(\alpha y)=0\}$ as a defining set , where $\alpha \in {\mathbb{F}}_{p^{s_2}}^{\ast }$ and $f(x)$ is a quadratic form over ${\mathbb{F}}_{p^{s_1}}$ with values in ${\text{F}}_p$, 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.