Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, secure two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and then determining their weight distributions have been interesting in coding theory and cryptography. In this paper, a generic construction for binary linear codes with dimension $m+2$ is presented, then a necessary and sufficient condition for this binary linear code to be minimal is derived. Based on this condition and exponential sums, a new class of minimal binary linear codes violating the Ashikhmin-Barg condition is obtained, and then their weight enumerators are determined.

翻译：最近,由于在秘密共享计划、安全的双方计算等应用中应用了最低线性代码,因此对最低线性代码进行了广泛研究。 建立违反Ashikhmin-Barg条件的最低限度线性代码,然后确定其重量分布,在编码理论和加密中很有意思。 在本文中,提出了一个尺寸为m+2美元的双线性代码的通用结构,然后得出了使这一双线性代码最小化的一个必要和充分的条件。 根据这一条件和指数数字,获得了一个新的违反Ashikhmin-Barg条件的最低限度双线性代码类别,然后确定了其重量计算器。