Weight hierarchies of a class of three-weight p-ary linear codes from inhomogeneous quadratic functions

The weight hierarchy of a linear code have been an important research topic in coding theory since Wei’s original work in 1991. In this paper, choosing \(D=\Big \{(x,y)\in \Big ({\mathbb {F}}_{p^{s_1}}\times {\mathbb {F}}_{p^{s_2}}\Big )\Big \backslash \{(0,0)\}: f(x)+\text {Tr}_1^{s_2}(\alpha y)=0\Big \}\) $D=\{(x,y)\in ({\mathbb{F}}_{p^{s_1}}\times {\mathbb{F}}_{p^{s_2}})\backslash \{(0,0)\}:f(x)+{\text{Tr}}_1^{s_2}(\alpha y)=0\}$ as a defining set, where \(\alpha \in {\mathbb {F}}_{p^{s_2}}^*\) $\alpha \in {\mathbb{F}}_{p^{s_2}}^{\ast }$ , f(x) is a quadratic form over \({\mathbb {F}}_{p^{s_1}}\) ${\mathbb{F}}_{p^{s_1}}$ with values in \({\mathbb {F}}_p\) ${\mathbb{F}}_p$ and f(x) can be non-degenerate or not, we construct a family of three-weight p-ary linear codes and determine their weight distributions and weight hierarchies completely. Most of the codes can be used in secret sharing schemes.