Thresholds of Random Quasi-Abelian Codes

Abstract

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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> ), 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.