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.

最小线性码构成一类特殊的线性码，在秘密共享和安全的两方计算中具有重要的应用。这些代码的特征是线性独立的代码字不会相互覆盖。分别用\（w _ {\ mathrm {min}} \）和\（w _ {\ mathrm {max}} \）表示二进制代码的最小和最大权重，当\（w_ {\ mathrm {min}} / w _ {\ mathrm {max}}> 1/2 \）（所谓的Ashikhmin–Barg的边界），而如果\（w _ {\ mathrm {min}} / w _ {\ mathrm {max}} \ le 1/2 \）。在本文中，我们扩展了Ding等人最初采取的措施。在[8]设计最小二进制线性码满足\（W _ {\ mathrm {分钟}} / W _ {\ mathrm {MAX}} \文件1/2 \） ，其被命名为宽在这篇文章。我们首先提出两种通用的方法来构造宽泛的最小二进制线性代码，这些方法使用一类常规的Maiorana-McFarland（\（\ mathcal {GMM} \））函数。第一种构造类似于Ding等人提出的构造。第二种结构类似于Mesnager等人最近提供的结构。[15]。但是，在某些情况下，我们的构造会产生具有更好的最小距离的代码。还提供了这些代码的确切重量分布。然后扩展这些方法，以增加代码的尺寸。线性代码的尺寸\（{\ mathcal {C}} _˚F\）从一个布尔函数导出˚F可提高邻接的码字\（{\ mathcal {C}} _ {D_ \伽马F} \），它指的是代码在方向\（\ gamma \）上与f的（合适的）导数关联。最值得注意的是，结合两个布尔函数的直接和与合适的导数子空间，我们获得了具有较大尺寸的最小代码。此外，当采用某些特殊类别的置换（例如AB（几乎弯曲）或反函数）时，这些较宽的最小代码具有较大的最小距离。