  Duadic codes are a class of cyclic codes that generalizes quadratic residue
codes from prime to composite lengths. For every prime power q, we characterize
the integers n such that over the finite field with q^2 elements there is a
duadic code of length n having an Hermitian self-dual parity-check extension.
  We derive using analytic number theory asymptotic estimates for the number of
such n as well as for the number of lengths for which duadic codes exist.
