  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.
