Refined bounds on the empirical distribution of good channel codes via concentration inequalities

We derive sharpened inequalities on the empirical output distribution of good channel codes with deterministic encoders and with non-vanishing maximal probability of decoding error. These inequalities refine recent bounds of Polyanskiy and Verdú by identifying closed-form expressions for certain asymptotic terms, which facilitates their calculation for finite blocklengths. The analysis relies on concentration-of-measure inequalities, specifically on McDiarmid´s method of bounded differences and its close ties to transportation inequalities for weighted Hamming metrics. An operational implication of the new bounds is addressed.