The lengths of Hermitian self-dual extended duadic codes

Duadic codes are a class of cyclic codes that generalize quadratic residue codes from prime to composite lengths. For every prime power qq, we characterize integers nn such that there is a duadic code of length nn over Fq2Fq2 with a Hermitian self-dual parity-check extension. We derive asymptotic estimates for the number of such nn as well as for the number of lengths for which duadic codes exist.