AI 中文总结
研究高信噪比下实加性高斯白噪声信道上随机球面码本，通过刻画错误概率转变规律得到集合可达数据速率的高SNR展开，分析其与逆界的关系及差距，确定了错误概率固定时差距随码长变化情况及相关可靠性缩放。
AI 中文摘要
本文刻画了在高信噪比（SNR）情况下，实加性高斯白噪声信道上的随机球面码本，此时码长固定且码本大小随SNR增长。在该情况下，随机球面集合呈现由球的内在维度和码本增长规模主导的尖锐错误概率转变。低于临界码本增长规模时，集合平均错误概率消失；在临界规模时，收敛到非平凡极限；高于该规模时，趋近于1。通过反转此转变规律，得到了对于规定错误概率的集合可达数据速率的高SNR展开。此速率与相应逆界具有相同的主导高SNR增长，可靠性通过常数阶项体现。随着SNR增加且错误概率固定时，集合可达速率与逆界之比趋于1，但它们的加性差异通常趋近于一个与码长和可靠性相关的正极限。我们刻画了这个作为码长和错误概率函数的极限速率界差距。对于每个固定错误概率，随着码长增加差距消失。我们进一步确定了在大码长极限下，错误概率降低阻止加性差距消失的可靠性缩放。
英文摘要
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.
