Minimal binary linear codes are a special class of binary codes with important applications in secret sharing and secure two-party computation. These codes are characterized by the property that none of the nonzero codewords is covered by any other codeword. Denoting by \(w_{{\min \limits }}\) and \(w_{{\max \limits }}\) the minimum and maximum weights of the codewords, respectively, such codes are relatively easy to design when the ratio \(w_{{\min \limits }}/w_{{\max \limits }}\) is larger than 1/2 (known as the Ashikhmin-Barg bound). On the other hand, a few known classes of minimal codes violate this bound, hence having the property \(w_{{\min \limits }}/w_{{\max \limits }} \leq 1/2\). In this article, we provide several explicit classes of minimal binary linear codes that violate the Ashikhmin-Barg bound while achieving a great variety of the ratio \(w_{{\min \limits }}/w_{{\max \limits }}\). Our first generic method employs suitable characteristic functions with relatively low weights within the range [n + 1,2^n− 2]. The second approach specifies characteristic functions with weights in [2^n− 2 + 1,2^n− 2 + 2^n− 3 − 1], whose supports contain a skewed (removing one element) affine subspace of dimension n − 2. Finally, we also characterize an infinite family of minimal codes based on the class of so-called root Boolean functions of weight 2^n− 1 − (n − 1), useful in specific hardware testing applications. Consequently, many infinite classes of minimal codes crossing the Ashikhmin-Barg bound are derived from an ample range of characteristic functions. In certain cases, we completely specify the weight distributions of the resulting codes.

最小二进制线性码是一类特殊的二进制码，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，任何其他代码字都不会覆盖任何非零代码字。用\（w _ {{\ min \ limits}} \）和\（w _ {{\ max \ limits}} \）分别表示码字的最小权重和最大权重。\（w _ {{\ min \ limits}} / w _ {{\ max \ limits}} \）大于1/2（称为Ashikhmin-Barg边界）。另一方面，一些已知的最小代码类违反了此限制，因此具有属性\（w _ {{\ min \ limits}} / w _ {{\ max \ limits}} \ leq 1/2 \）。在本文中，我们提供了几个显式类的最小二进制线性代码，它们违反Ashikhmin-Barg界限，同时实现多种比率\（w _ {{\ min \ limits}} / w _ {{\\ max \ limits}} \）。我们的第一个通用方法采用合适的特征函数，其权重在[ n + 1,2 ^{n -2} ]范围内相对较低。第二种方法指定权重为[2 ^{n -2} + 1,2 ^{n -2} + 2 ^{n -} 3-1]的特征函数，其支持包含维度为n的倾斜（移除一个元素）仿射子空间。− 2.最后，我们还基于权重为2 ^{n − 1} −（n − 1）的所谓根布尔函数的类别，描述了无限的最小代码系列，这些特征在特定的硬件测试应用中很有用。因此，从丰富的特征函数范围中得出了许多跨越Ashikhmin-Barg边界的最小代码的无限类。在某些情况下，我们完全指定结果代码的权重分布。