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={(x,y)∈ (𝔽_p^s_1×𝔽_p^s_2 )\{(0,0)}: f(x)+Tr_1^s_2(α y)=0} as a defining set, where α∈𝔽_p^s_2^* , f ( x ) is a quadratic form over 𝔽_p^s_1 with values in 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.