The lengths of Hermitian Self-Dual Extended Duadic Codes

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.Comment: To appear in the Journal of Pure and Applied Algebra. 21 pages and 1 Table. Corollary 4.9 and Theorem 5.8 have been added. Some small changes have been mad

Around Pelikan's conjecture on very odd sequences

Very odd sequences were introduced in 1973 by J. Pelikan who conjectured that there were none of length >=5. This conjecture was disproved by MacWilliams and Odlyzko in 1977 who proved there are in fact many very odd sequences. We give connections of these sequences with duadic codes, cyclic difference sets, levels (Stufen) of cyclotomic fields, and derive some new asymptotic results on their lengths and on S(n), which denotes the number of very odd sequences of length n.Comment: 21 pages, two tables. Revised version with improved presentation and correction of some typos and minor errors that will appear in Manuscripta Mathematic