  We study the size (or volume) of balls in the metric space of permutations,
$S_n$, under the infinity metric. We focus on the regime of balls with radius
$r = \rho \cdot (n\!-\!1)$, $\rho \in [0,1]$, i.e., a radius that is a constant
fraction of the maximum possible distance. We provide new lower bounds on the
size of such balls. These new lower bounds reduce the asymptotic gap to the
known upper bounds to at most $0.029$ bits per symbol. Additionally, they imply
an improved ball-packing bound for error-correcting codes, and an improved
upper bound on the size of optimal covering codes.
