A class of self-dual quasi-abelian codes of index 2 over any finite field$F$is introduced. By counting the number of such codes and the number of the codes in this class whose relative minimum weights are small, such codes are proved to be asymptotically good provided −1 is a square in$F$. Moreover, a class of self-orthogonal quasi-abelian codes of index 2 is defined; and such codes always exist. In a way similar to that for self-dual quasi-abelian codes of index 2, it is proved that these self-orthogonal quasi-abelian codes of index 2 are asymptotically good.

Bazzi and Mitter showed that binary dihedral group codes are asymptotically good. In this paper we prove that the dihedral group codes over any finite field with strong duality property are asymptotically good. If the characteristic of the field is even, self-dual dihedral group codes are asymptotically good. If the characteristic of the field is odd, maximal self-orthogonal dihedral group codes and LCD dihedral group codes are asymptotically good.

Symbol-pair codes introduced by Cassuto and Blaum (2010) are designed to protect against pair errors in symbol-pair read channels. The higher the minimum pair distance, the more pair errors the code can correct. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense that pair distance cannot be improved for given length and code size. The contribution of this paper is twofold. First, we present three lower bounds for the minimum pair distance of constacyclic codes, the first two of which generalize the previously known results due to Cassuto and Blaum (2011) and Kai et al. (2015). The third one exhibits a lower bound for the minimum pair distance of repeated-root cyclic codes. Second, we obtain new MDS symbol-pair codes with minimum pair distance seven and eight through repeated-root cyclic codes.

For a q-ary random quasi-Abelian code with fixed coindex and constant rate r, it is shown that the Gilbert-Varshamov (GV)-bound is a threshold point: if r is less than the GV-bound at δ ∈ (0, 1 - q-1), then the probability of the relative distance of the random code being greater than δ approaches 1 as the index goes to infinity; whereas, if r is bigger than the GV-bound at δ, then the probability approaches 0. As a corollary, there exist numerous asymptotically good quasi-Abelian codes attaining the GV-bound.