Random Spherical Codes at High SNR: Error Transitions, Fixed-Error Data Rates, and Converse Gaps
Information Theory
Summary
The authors study how well random spherical codebooks perform when sending data over a noisy channel at very high signal strength, with fixed message length and increasing code size. They find a sharp change in error rates depending on how quickly the code size grows, revealing when errors fade away or become almost certain. Using this, they describe the maximum data rate achievable for a given error chance, showing it closely matches theoretical limits as the signal strength grows. However, a small difference remains that depends on message length and reliability, which disappears as message length increases unless the required error gets even smaller. In short, the paper clarifies how error probability and message length influence data rates in this high-signal scenario.
Authors
Nikola Zlatanov
Abstract
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.