Beyond Minimum Distance: The Optimal Leading Coefficient in the High-SNR Error-Probability Expansion for AWGN Spherical Codes

2026-08-26

AI Summary

该研究探讨了通过在每个信噪比下重新优化码本，以减小高信噪比下的错误概率的领先系数，从而提高AWGN球面码性能的方法。

Abstract

Packing-optimal $(M,n)$ spherical codes attain the largest achievable minimum distance and hence the optimal error exponent at high SNR. Among these codebooks held fixed as SNR grows, the smallest leading coefficient, $K_{M,n}^{\mathrm{fix}}$, equals the smallest number of ordered closest pairs. We show that reoptimizing the codebook at every SNR can yield a smaller leading coefficient $K_{M,n}^\ast$. Specifically, we show that the SNR-wise minimum exact maximum-likelihood error probability is $P_e^\ast=B_γ^\ast(K_{M,n}^\ast+o(1))$, while the minimum exact error over packing-optimal codebooks is $P_e^{\mathrm{fix}}=B_γ^\ast(K_{M,n}^{\mathrm{fix}}+o(1))$, where $B_γ^\ast$ is a common factor. We prove that $K_{M,n}^\ast\leq K_{M,n}^{\mathrm{fix}}$. Strict inequality, $K_{M,n}^\ast < K_{M,n}^{\mathrm{fix}}$ therefore gives a smaller exact error at all sufficiently high SNRs, i.e., $P_e^\ast<P_e^{\mathrm{fix}}$. We also characterize $K_{M,n}^\ast$ as the limit of the minimum Gaussian soft-packing energy. An orthoplex-bound construction explains strict inequality: an SNR-dependent perturbation makes one class of limiting closest pairs slightly closer, without changing the optimal exponent or their unit contributions to the constructed family's leading coefficient, while moving the remaining closest pairs farther apart on a larger but still vanishing scale, causing their contributions to vanish. For $2\leq M\leq n+1$, we prove $K_{M,n}^\ast=K_{M,n}^{\mathrm{fix}}=M(M-1)$. For $M=n+k$, $2\leq k\leq n$, we prove $K_{n+k,n}^{\mathrm{fix}}=4n(k-1)$ and $K_{n+k,n}^\ast\leq 4k(k-1)$. In particular, $K_{n+2,n}^\ast=8<4n=K_{n+2,n}^{\mathrm{fix}}$ for $n\geq3$, so SNR-wise optimization improves the asymptotic error by the factor $n/2$. At $k=n$, both coefficients equal $4n(n-1)$. For $3\leq k\leq n-1$, we conjecture $K_{n+k,n}^\ast=4k(k-1)$.

Problem

• High-SNR
• Error-Probability
• Spherical Codes
• Leading Coefficient
• Packing-Optimal

Innovation

• SNR-wise optimization
• leading coefficient
• spherical codes
• error probability
• orthoplex-bound construction