A new mathematical study finds that a signal codebook can do better at very high signal-to-noise ratios when its points are allowed to move with the noise level. In the nonterminal orthoplex range, the paper constructs such a moving family with a lower leading error coefficient than the best fixed codebook.
The result comes with an important boundary. For codebooks arranged as regular simplices, the moving and fixed designs have the same leading coefficient. In the intermediate part of the orthoplex range, the paper proves an upper bound but labels equality with that bound a conjecture.
The constant behind the error curve
The work studies finite spherical codebooks: labeled sets of unit-length signal vectors. It is a modeling paper, with no empirical participant sample or experimental dataset. The question is how small the remaining leading coefficient can be when minimum message error is optimized in the high-SNR limit.
For a codebook held fixed, the high-SNR calculation is governed by its closest pairs. More precisely, the leading coefficient is the number of ordered pairs of codewords that sit at the minimum distance. The paper also introduces a Gaussian soft-packing energy, a score for how tightly codewords are packed, and shows that minimizing it for SNR-dependent codebooks is asymptotically equivalent to minimizing the normalized exact maximum-likelihood error.
The general result says that, for every fixed finite codebook size and dimension in the stated positive-separation setting, the SNR-wise optimum and the minimum soft-packing energy approach the same finite value, called K*. That value lies between 2 and M(M − 1), where M is the number of codewords.
This SNR-wise optimum can never have a larger leading coefficient than the best packing-optimal codebook kept fixed. If K* is strictly smaller than the fixed benchmark, the optimized exact error is eventually smaller at high SNR, and the ratio of the two errors approaches K*/K_fix.
The clearest gap appears in orthoplex designs
The sharpest contrast appears in a class of spherical-packing problems known as the orthoplex-bound range. Here the number of codewords is written as M = n + k, with 2 ≤ k ≤ n. The best fixed design has leading coefficient K_fix = 4n(k − 1), achieved by combining orthogonal antipodal pairs with a regular simplex.
The authors construct an SNR-dependent family whose leading coefficient is 4k(k − 1) in the nonterminal part of that range. This proves that K* is no larger than 4k(k − 1), and gives a moving-family error scale that is k/n of the fixed benchmark.
At the left endpoint, k = 2, the result is exact: K* = 8, while the best fixed coefficient is K_fix = 4n. For dimensions n ≥ 3, the fixed benchmark is therefore n/2 times the moving family's asymptotic error scale.
At the other endpoint, k = n, the full cross-polytope is optimal for the soft-packing problem at every positive SNR. The resulting high-SNR coefficients coincide at K* = K_fix = 4n(n − 1), so there is no leading-order advantage from moving the codebook there.
Between those endpoints, for 3 ≤ k ≤ n − 1, the paper does not prove that 4k(k − 1) is the exact optimum. It records that equality as a conjecture, leaving open whether the constructed family is already sharp.
A carefully scaled path toward the packing
The proposed mechanism is more subtle than simply spreading every pair as far apart as possible. Families that remain competitive on the high-SNR packing scale are forced toward the set of packing-optimal configurations, but their approach can still be tuned. Among pairs that are closest in the limiting packing, the correlation path on the 1/(nγ) scale determines how much each pair contributes to the final coefficient.
That contribution need not be an ordinary whole-number pair count. Depending on the SNR-scaled path, a limiting closest pair can contribute nothing, or a fractional, unit-sized, or greater-than-unit amount. The construction uses this freedom across several vanishing scales: some limiting closest pairs are made slightly worse on a negligible scale, while others are separated on a larger vanishing scale and contribute zero.
The paper's conclusions are mathematical statements about finite unit-norm spherical codebooks and maximum-likelihood decoding, rather than evidence from an empirical sample. Because the soft-packing equivalence is asymptotic, the headline result concerns the high-SNR limit.
The supplied front matter identifies the work as arXiv:2608.25805v1 in cs.IT, dated 26 August 2026. Funding is not reported in the supplied text.
Paper data and sources
Original title: Beyond Minimum Distance: The Optimal Leading Coefficient in the High-SNR Error-Probability Expansion for AWGN Spherical Codes
Authors: Nikola Zlatanov
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text