Thresholds of Random Quasi-Abelian 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.