Weight distributions of several families of 3-weight binary linear codes

The linear codes with a few weights have been applied widely in combinatorial designs, secret sharing, association schemes, authentication codes and strongly regular graphs. In this paper, we first correct an erroneous result about the exponential sum $\sum_{x\in \mathbb{F}_{2^{e}}}\chi_1\left(ax^{2^{\alpha}+1}+bx\right).$ Then, using the above exponential sum, we construct several families of binary linear codes of $3$-weight and determine their weight distributions. Moreover, Most of them can be used in secret sharing schemes.