  Volume estimates of metric balls in manifolds find diverse applications in
information and coding theory. In this paper, some new results for the volume
of a metric ball in unitary group are derived via various tools from random
matrix theory. The first result is an integral representation of the exact
volume, which involves a Toeplitz determinant of Bessel functions. The
connection to matrix-variate hypergeometric functions and Szeg\H{o}'s strong
limit theorem lead independently from the finite size formula to an asymptotic
one. The convergence of the limiting formula is exceptionally fast due to an
underlying mock-Gaussian behavior. The proposed volume estimate enables simple
but accurate analytical evaluation of coding-theoretic bounds of unitary codes.
In particular, the Gilbert-Varshamov lower bound and the Hamming upper bound on
cardinality as well as the resulting bounds on code rate and minimum distance
are derived. Moreover, bounds on the scaling law of code rate are found.
Lastly, a closed-form bound on diversity sum relevant to unitary space-time
codes is obtained, which was only computed numerically in literature.
