Wide minimal binary linear codes from the general Maiorana-McFarland class
Wide minimal binary linear codes from the general Maiorana-McFarland class
复制标题
来自一般 Maiorana-McFarland 类的宽最小二进制线性码
DOI：
10.1007/s10623-021-00883-7
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发表时间：
2021
期刊：
影响因子：
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通讯作者：
Yongzhuang Wei
中科院分区：
文献类型：
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作者：
Fengrong Zhang;Enes Pasalic;Rene Rodriguez;Yongzhuang Wei
最小线性码形成了一类特殊的线性码，在秘密共享和安全双方计算中有重要的应用。这些码的特征在于线性独立的码字不相互覆盖的特性。用和分别表示二进制码的最小和最大重量，当(所谓的Ashikhmin-Barg‘s界)时，这样的码可以相对容易地设计，而当时，它们的构造变得更难。在本文中，我们将Dinget等人最初提出的方法扩展到设计满足的最小二进制线性码，这在本文中称为Wide。我们首先提出了两种利用一类广义Maiorana-McFarland()函数构造广义极小二进制线性码的一般方法。第一种结构类似于Ding等人提出的结构。第二种结构类似于Mesnager等人最近提供的结构。。然而，在某些情况下，我们的构造产生的代码具有更好的最小距离。文中还给出了这些码的精确重量分布。然后对这些方法进行扩展，以增加码的维度。从布尔函数f导出的线性代码的维度可以通过邻接代码字来增加，该代码字指的是与(适当的)方向上的导数相关联的代码。最值得注意的是，结合两个布尔函数的直和和适当的导数子空间，我们得到了具有相当大的维度的宽极小码。此外，当使用某些特殊的置换类别，例如AB(几乎弯曲的)或反函数时，这些宽的极小码具有很大的最小距离。
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 byandthe minimum and maximum weights of a binary code, respectively, such codes can be designed relatively easy when(the so-called Ashikhmin–Barg’s bound), whereas their construction becomes harder if. In this article, we extend the initiative originally taken by Dinget al.in to design minimal binary linear codes that satisfy, which are namedwidein this article. We first propose two generic methods for constructing wide minimal binary linear codes that use a class of general Maiorana-McFarland () 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. . 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 codederived from a Boolean functionfcan be increased by adjoining the codewords of, which refers to the code associated to a (suitable) derivative offat direction. 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.