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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 constantfraction of the maximum possible distance. We provide new lower bounds on thesize of such balls. These new lower bounds reduce the asymptotic gap to theknown upper bounds to at most $0.029$ bits per symbol. Additionally, they implyan improved ball-packing bound for error-correcting codes, and an improvedupper bound on the size of optimal covering codes.

Improved Lower Bounds on the Size of Balls over Permutations with the Infinity Metric