  A permutation array(or code) of length $n$ and distance $d$, denoted by
$(n,d)$ PA, is a set of permutations $C$ from some fixed set of $n$ elements
such that the Hamming distance between distinct members
$\mathbf{x},\mathbf{y}\in C$ is at least $d$. Let $P(n,d)$ denote the maximum
size of an $(n,d)$ PA. New upper bounds on $P(n,d)$ are given. For constant
$\alpha,\beta$ satisfying certain conditions, whenever $d=\beta n^{\alpha}$,
the new upper bounds are asymptotically better than the previous ones.
