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

This paper characterizes random spherical codebooks over the real additive white Gaussian noise channel in the high signal-to-noise ratio (SNR) regime when the blocklength is fixed and the codebook size grows with SNR.

In this regime, the random spherical ensemble exhibits a sharp error-probability transition governed by the intrinsic dimension of the sphere and the codebook-growth scale.

Below the critical codebook-growth scale, the ensemble-average error probability vanishes; at the critical scale, it converges to a nontrivial limit; and above that scale, it approaches one.

By inverting this transition law, we obtain the high-SNR expansion of the ensemble-achievable data rate for a prescribed error probability.

This rate has the same leading high-SNR growth as the corresponding converse rate bound, while reliability enters through the constant-order terms.

Consequently, the ratio of the ensemble-achievable rate to the converse rate bound tends to one as the SNR increases.

Their additive difference, however, generally approaches a positive blocklength- and reliability-dependent limit.

We characterize this limiting rate-bound gap jointly as a function of blocklength and error probability.

For every fixed error probability, the gap vanishes as the blocklength increases.

We further identify the reliability scalings under which a decreasing error probability prevents the additive gap from vanishing in the large-blocklength limit.