Linear codes play a crucial role in various fields of engineering and mathematics, including data storage, communication, cryptography, and combinatorics. Minimal linear codes, a subset of linear codes, are particularly essential for designing effective secret sharing schemes. In this paper, we introduce several classes of minimal binary linear codes by carefully selecting appropriate Boolean functions. These functions belong to a renowned class of Boolean functions, namely, the general Maiorana–McFarland class. We employ a method first proposed by Ding et al. (IEEE Trans Inf Theory 64(10):6536–6545, 2018) to construct minimal codes violating the Ashikhmin–Barg bound (wide minimal codes) by using Krawtchouk polynomials. The lengths, dimensions, and weight distributions of the obtained codes are determined using the Walsh spectrum distribution of the chosen Boolean functions. Our findings demonstrate that a vast majority of the newly constructed codes are wide minimal. Furthermore, our proposed codes exhibit a significantly larger minimum distance, in some cases, compared to some existing similar constructions. Finally, we address this method, based on Krawtchouk polynomials, more generally, and highlight certain generic properties related to it. These general results offer insights into the scope of this approach.

线性码在工程和数学的各个领域中发挥着至关重要的作用，包括数据存储、通信、密码学和组合学。最小线性码是线性码的子集，对于设计有效的秘密共享方案尤其重要。在本文中，我们通过仔细选择适当的布尔函数来介绍几类最小二进制线性码。这些函数属于著名的布尔函数类，即一般的 Maiorana-McFarland 类。我们采用了 Ding 等人首先提出的方法。(IEEE Trans Inf Theory 64(10):6536–6545, 2018）使用 Krawtchouk 多项式构造违反 Ashikhmin–Barg 界限的最小代码（宽最小代码）。所获得代码的长度、尺寸和重量分布是使用所选布尔函数的沃尔什谱分布来确定的。我们的研究结果表明，绝大多数新构建的代码都是极简的。此外，在某些情况下，与一些现有的类似结构相比，我们提出的代码表现出明显更大的最小距离。最后，我们更一般地基于 Krawtchouk 多项式来讨论该方法，并强调与其相关的某些通用属性。这些一般结果提供了对该方法范围的见解。