Generalized hamming weights of linear codes from cryptographic functions

Abstract

<p style='text-indent:20px;'>The generalized Hamming weights of linear codes have attracted scholars' attention since Wei used them to characterize the cryptography performance of a linear code over the wire-tap channel of type II in 1991. Generally speaking, it is hard to determine linear codes' generalized Hamming weights, especially the weight hierarchy. On the other hand, since Ding and Niederreiter presented a generic approach to construct linear codes with good properties by suitable defining sets in 2007, many linear codes with few weights have been obtained using cryptographic functions. However, there does not seem to be much research on the generalized Hamming weights of linear codes from cryptographic functions.</p><p style='text-indent:20px;'>In this paper, we first provide a general formula to compute the generalized Hamming weights of linear codes from the defining sets, which generalizes several known results. Next, some weight hierarchies of linear codes from three famous classes of Bent functions (the Maiorana-McFarland class, the class <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{PS}_{ap} $\end{document}</tex-math></inline-formula>, the class <inline-formula><tex-math id="M2">\begin{document}$ H $\end{document}</tex-math></inline-formula>) and two classes of Semi-Bent functions (the Maiorana-McFarland class, the summation of <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{PS}_{ap} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ H $\end{document}</tex-math></inline-formula>) are considered. We hope this paper can attract interested scholars to find more results about the generalized Hamming weights of linear codes from cryptographic functions.</p>