Concentration Properties of Random Codes

This paper shows that, for discrete memoryless channels, the error exponent of a randomly generated code with independent codewords converges in probability to its expectation—the typical error exponent. For high rates, the result follows from the fact that the random-coding error exponent and the sphere-packing error exponent coincide. For low rates, instead, the convergence is based on the fact that the union bound accurately characterizes the error probability. The paper also zooms into the behavior at asymptotically low rates, and shows that the normalized error exponent converges in distribution to the standard Gaussian or a Gaussian-like distribution. We also state several results on the convergence of the error probability and error exponent for generic ensembles and channels.