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

2026-07-16

AI Summary

This work investigates the error performance and achievable rates of random spherical codes over the real-valued additive white Gaussian noise channel in the high signal-to-noise ratio (SNR) regime. By employing ensemble analysis of random codes, high-SNR asymptotic expansions, and converse bounding techniques, the study reveals a three-phase transition behavior in the average error probability as the codebook size increases. Building on this characterization, it jointly quantifies—for the first time—the impact of blocklength and target reliability on the additive gap between achievable rates and converse bounds. This gap converges to a positive value dependent on both blocklength and target error probability, yet vanishes as blocklength grows; concurrently, the ratio of achievable rate to converse bound approaches unity, demonstrating their asymptotic tightness at high SNR.

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.

Problem

• random spherical codes
• high SNR
• error probability
• data rate
• converse gap

Innovation

• random spherical codes
• high-SNR analysis
• error probability transition
• achievable rate
• converse gap