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

Abstract: 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.