  Generalized Galois numbers count the number of flags in vector spaces over
finite fields. Asymptotically, as the dimension of the vector space becomes
large, we give their exponential growth and determine their initial values. The
initial values are expressed analytically in terms of theta functions and
Euler's generating function for the partition numbers. Our asymptotic
enumeration method is based on a Demazure module limit construction for
integrable highest weight representations of affine Kac-Moody algebras. For the
classical Galois numbers, that count the number of subspaces in vector spaces
over finite fields, the theta functions are Jacobi theta functions. We apply
our findings to the asymptotic number of q-ary codes, and conclude with some
final remarks about possible future research concerning asymptotic enumerations
via limit constructions for affine Kac-Moody algebras.
