The Error Exponent of Generalized Random-Gilbert Varshamov Codes

We introduce a random code construction for channel coding in which the codewords are constrained to be well-separated according to a given distance function, analogously to an existing construction attaining the Gilbert-Varshamov bound. We derive an achievable error exponent for this construction, and prove its tightness with respect to the ensemble average. We show that the exponent recovers the Csiszár and Körner exponent as a special case by choosing the distance function to be the negative of the empirical mutual information. We further establish the optimality of this distance function with respect to the exponent of the random coding scheme.