  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. This correspondence focuses on the lower bound on
$P(n,d)$. First we give three improvements over the Gilbert-Varshamov lower
bounds on $P(n,d)$ by applying the graph theorem framework presented by Jiang
and Vardy. Next we show another two new improved bounds by considering the
covered balls intersections. Finally some new lower bounds for certain values
of $n$ and $d$ are given.
