In this paper, three classes of binary linear codes with few weights are proposed from vectorial Boolean power functions, and their weight distributions are completely determined by solving certain equations over finite fields. In particular, a class of simplex codes and a class of first-order Reed-Muller codes can be obtained from our construction by taking the identity map, whose dual codes are Hamming codes and extended Hamming codes, respectively.

中文翻译：

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向量布尔幂函数的线性代码

本文从矢量布尔幂函数中提出了三类权重较小的二进制线性码，并通过求解有限域上的某些方程来完全确定它们的权重分布。特别地，通过采用身份映射可以从我们的构造中获得一类单纯形码和一阶Reed-Muller码，其对偶图分别是汉明码和扩展汉明码。