  In this paper, we study the bound on three kinds of hash family using the
Singleton bound. To $\epsilon-U(N; n, m)$ hash family, in the caes of $n>m^2>1$
and $1\geq\epsilon\geq \epsilon_1(n, m)$, we get that the new bound is better.
To $\epsilon-\bigtriangleup U(N; n, m)$ hash family, in the case of $n>m>1$ and
$1\geq\epsilon\geq\epsilon_3(n,m)$, the new bound is better. To $\epsilon-SU(N;
n, m)$ hash family, in the case of $n>2^m>2$ and $1\geq\epsilon\geq
\epsilon_4(n, m)$, we get that the new bound is better.
