Linear codes with few weights have applications in data storage systems, secret sharing schemes and authentication codes. In this paper, inspired by the butterfly structure [6], [29] and the works of Li, Yue and Fu [21] and Jian, Lin and Feng [19], we introduce a new defining set with the form of the closed butterfly structure and consequently we obtain three classes of 3-weight binary linear codes and a class of 4-weight binary linear codes whose dual is optimal. The lengths and weight distributions of these four classes of linear codes are completely determined by some detailed calculations on certain exponential sums. Computer experiments show that many (almost) optimal codes can be obtained from our construction.

具有少量权重的线性代码在数据存储系统、秘密共享方案和认证代码中具有应用。在本文中，受蝴蝶结构 [6]、[29] 以及 Li、Yue 和 Fu [21] 以及Jian、Lin 和 Feng [19] 的作品的启发，我们引入了一个新的定义集，其形式为闭式蝴蝶结构，因此我们得到三类3权重二进制线性码和一类对偶最优的4权重二进制线性码。这四类线性码的长度和权重分布完全由对某些指数和的一些详细计算确定。计算机实验表明，可以从我们的构造中获得许多（几乎）最优代码。