Boolean functions have very nice applications in coding theory and cryptography. In coding theory, Boolean functions have been used to construct linear codes in different ways. The objective of this paper is to construct binary linear codes with few weights using the defining-set approach. The defining sets of the codes presented in this paper are defined by some special Boolean functions and some additional restrictions. First, two families of binary linear codes with at most three or four weights from Boolean functions with at most three Walsh transform values are constructed and the parameters of their duals are also determined. Then several classes of binary linear codes with explicit weight enumerators are produced. Some of the binary linear codes are optimal or almost optimal according to the tables of best codes known maintained at http://www.codetables.de, and the duals of some of them are distance-optimal with respect to the sphere packing bound.

布尔函数在编码理论和密码学中有很好的应用。在编码理论中，布尔函数已被用于以不同方式构造线性代码。本文的目的是使用定义集方法构造具有很少权重的二进制线性代码。本文中给出的代码定义集是由一些特殊的布尔函数和一些附加限制定义的。首先，从具有最多三个沃尔什变换值的布尔函数中构造最多三个或四个权重的两个二进制线性码族，并确定它们的对偶参数。然后产生几类具有显式权重枚举器的二进制线性码。根据 http 中维护的已知最佳代码表，一些二进制线性代码是最佳的或几乎是最佳的：