Concentration Properties of Generalized Random Gilbert-Varshamov Codes

We study the typical error exponent of constant composition generalized random Gilbert-Varshamov (RGV) codes over discrete memoryless channels (DMC) channels with generalized likelihood decoding. We show that the typical error exponent of the RGV ensemble is equal to the expurgated error exponent, provided that the RGV codebook parameters are chosen appropriately. We also prove that the exponent of a randomly chosen RGV code converges in probability to the typical error exponent; the lower tail is shown to decay exponentially while the upper tail decays double-exponentially above the expurgated exponent.