Random Spherical Codes at High SNR: Error Transitions, Fixed-Error Data Rates, and Converse Gaps

Abstract

This paper characterizes random spherical codebooks over the real additive white Gaussian noise channel in the high signal-to-noise ratio (SNR) regime in which the blocklength is fixed, the SNR per real channel use tends to infinity, and the codebook size grows with SNR. The ensemble exhibits a sharp error-probability transition governed by the intrinsic dimension of the sphere and the codebook-growth order. Below the critical codebook-growth order, the ensemble-average error probability vanishes; at the critical order, it converges to a nontrivial limit; and above that order, it approaches one. For a fixed target average error probability, this transition yields the high-SNR expansion of the ensemble-achievable data rate. The achievable rate and the corresponding converse rate bound have the same high-SNR prelog, establishing first-order optimality within the deterministic equal-energy, average-error class. Their ratio tends to one for every fixed blocklength and target error probability, but their additive difference approaches a strictly positive limit that depends on both. We characterize this rate-bound gap jointly in blocklength and error probability. At fixed error probability, it vanishes with increasing blocklength, with a universal leading $1/n$ behavior and reliability dependence first appearing at the next order. For error probabilities that decrease exponentially with blocklength, we identify the threshold between vanishing and nonvanishing limiting gaps. We also derive blocklength laws for meeting a prescribed gap as the target error probability becomes more stringent.