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For a long time, asymptotically good self-dual codes have been known to exist. Asymptotically good 2-quasicyclic codes of rate 1/2 have also been known to exist for a long time. Recently, it was proved that there are binary self-dual n/3-quasicyclic codes of length n asymptotically meeting the Gilbert-Varshamov bound. Unlike 2-quasicyclic codes, which are defined to have a cyclic group of order n/2 as a subgroup of their permutation group, the n/3-quasicyclic c codes are defined with a permutation group of fixed order of 3. So, from the decoding point of view, 2-quasicyclic c codes are preferable to n/3-quasicyclic c codes. In this correspondence, with the assumption that there are infinite primes p with respect to (w r t.) which 2 is primitive, we prove that there exist classes of self-dual 2p-quasicyclic c codes and Type II 8p-quasicyclic c codes of length respectively 2p2 and 8p2 which asymptotically meet the Gilbert-Varshamov bound. When compared with the order of the defining permutation groups, these classes of codes lie between the 2-quasicyclic c codes and the n/3-quasicyclic c codes of length n, considered in previous works.