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

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, $K_{M,n}<sup>{\mathrm{fix}}$, equals the smallest number of ordered closest pairs. We show that reoptimizing the codebook at every SNR can yield a smaller leading coefficient $K_{M,n}<sup>\ast$. Specifically, we show that the SNR-wise minimum exact maximum-likelihood error probability is $P_e<sup>\ast=B_γ<sup>\ast(K_{M,n}<sup>\ast+o(1))$, while the minimum exact error over packing-optimal codebooks is $$P_e<sup>{\mathrm{fix}}=B_γ<sup>\ast(K_{M,n}<sup>{\mathrm{fix}}+o(1))$$, where $B_γ<sup>\ast$ is a common factor. We prove that $$K_{M,n}^\ast\leq K_{M,n}<sup>{\mathrm{fix}}$$. Strict inequality, $K_{M,n}^\ast &lt; K_{M,n}<sup>{\mathrm{fix}}$ therefore gives a smaller exact error at all sufficiently high SNRs, i.e., $P_e<sup>\ast&lt;P_e<sup>{\mathrm{fix}}$. We also characterize $K_{M,n}<sup>\ast$ 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 $K_{M,n}<sup>\ast=K_{M,n}<sup>{\mathrm{fix}}=M(M-1)$. For $M=n+k$, $2\leq k\leq n$, we prove $K_{n+k,n}<sup>{\mathrm{fix}}=4n(k-1)$ and $K_{n+k,n}^\ast\leq 4k(k-1)$. In particular, $K_{n+2,n}<sup>\ast=8&lt;4n=K_{n+2,n}<sup>{\mathrm{fix}}$ 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 $K_{n+k,n}<sup>\ast=4k(k-1)$.