Given positive integers $ n $ and $ d $, let $ M (n, d) $ denote the maximum size of a
permutation code of length $ n $ and minimum Hamming distance $ d $. The Gilbert-
Varshamov bound asserts that $ M (n, d)\geq n!/V (n, d-1) $ where $ V (n, d) $ is the volume
of a Hamming sphere of radius $ d $ in $\S_n $. Recently, Gao, Yang, and Ge showed that
this bound can be improved by a factor $\Omega (\log n) $, when $ d $ is fixed and $
n\to\infty $. Herein, we consider the situation where the ratio $ d/n $ is fixed and improve the …