Wide minimal binary linear codes from the general Maiorana–McFarland class

Abstract

Minimal linear codes form a special class of linear codes that have important applications in secret sharing and secure two-party computation. These codes are characterized by the property that linearly independent codewords do not cover each other. Denoting by $w_{\mathrm {min}}$ and $w_{\mathrm {max}}$ the minimum and maximum weights of a binary code, respectively, such codes can be designed relatively easy when $w_{\mathrm {min}}/w_{\mathrm {max}} > 1/2$ (the so-called Ashikhmin–Barg’s bound), whereas their construction becomes harder if $w_{\mathrm {min}}/w_{\mathrm {max}} \le 1/2$. In this article, we extend the initiative originally taken by Ding et al. in [8] to design minimal binary linear codes that satisfy $w_{\mathrm {min}}/w_{\mathrm {max}} \le 1/2$, which are named wide in this article. We first propose two generic methods for constructing wide minimal binary linear codes that use a class of general Maiorana-McFarland ($\mathcal {GMM}$) functions. The first construction is similar to the one proposed by Ding et al. and the second construction is similar to the one recently provided by Mesnager et al. [15]. Nevertheless, our constructions yield codes with better minimum distances in certain cases. The exact weight distributions of these codes are also provided. These approaches are then extended so that the dimension of the codes is increased. The dimension of the linear code ${\mathcal {C}}_f$ derived from a Boolean function f can be increased by adjoining the codewords of ${\mathcal {C}}_{D_\gamma f}$, which refers to the code associated to a (suitable) derivative of f at direction $\gamma$. Most notably, combining the direct sum of two Boolean functions and a suitable subspace of derivatives, we obtain wide minimal codes with a substantial larger dimension. Furthermore, these wide minimal codes feature a large minimum distance when employing some special classes of permutations, such as AB (almost bent) or the inverse function.