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There is the long-standing question whether the class of cyclic codes is asymptotically good. By an old result of Lin and Weldon, long Bose-Chaudhuri-Hocquenhem (BCH) codes are asymptotically bad. Berman proved that cyclic codes are asymptotically bad if only finitely many primes are involved in the lengths of the codes. We investigate further classes of cyclic codes which also turn out to be asymptotically bad. Based on reduction arguments we give some evidence that there are asymptotically good sequences of binary cyclic codes in which all lengths are prime numbers provided there is any asymptotically good sequence of binary cyclic codes.