We first establish a simple yet powerful necessary and sufficient condition for a binary linear code to be SO, leading to a complete characterization of singly-even codes in this family. We further derive necessary and sufficient conditions on Boolean and vectorial Boolean functions for generating such codes via a standard construction method. Building on this foundation, we propose three general frameworks for constructing binary SO singly-even minimal non-AB linear codes with few weights. The first two approaches are based on designing Boolean and vectorial Boolean functions that simultaneously satisfy multiple conditions. The third method generates new SO codes from existing ones. As a result, we obtain many infinite classes of binary self-orthogonal singly-even minimal linear codes violating the AB condition with few weights and fully determined weight distributions. Particularly, numerical results show that some duals of our codes are optimal or near-optimal.

翻译：我们首先建立了一个简单而强大的充要条件，用于判定二进制线性码是否为自正交码，从而完全刻画了该族中单偶码的特征。进一步，我们推导了布尔函数与向量布尔函数通过标准构造方法生成此类码的充要条件。在此基础上，我们提出了三种通用框架，用于构造违反AB条件的、少重量的二进制自正交单偶极小线性码。前两种方法基于设计同时满足多重条件的布尔函数与向量布尔函数。第三种方法从现有码生成新的自正交码。由此，我们获得了许多无限类的、违反AB条件的、具有少重量且权重分布完全确定的二进制自正交单偶极小线性码。特别地，数值结果表明，我们部分码的对偶码是最优或接近最优的。