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Beyond Minimum Distance: The Optimal Leading Coefficient in the High-SNR Error-Probability Expansion for AWGN Spherical Codes
Nikola Zlatanov
TL;DR
The paper asks whether packing-optimal spherical codes also optimize the high-SNR error coefficient when codewords may change with SNR. It introduces Gaussian soft-packing energy to characterize the SNR-wise optimum and proves that reoptimization can strictly improve the coefficient, with equality in the simplex range and explicit improvements in the orthoplex-bound range.
Problem
Packing determines the optimal high-SNR error exponent, but it is not known whether it determines the optimal leading coefficient when the codebook is reoptimized at every SNR.
Method
The paper transfers normalized exact ML error to a Gaussian soft-packing energy and analyzes SNR-dependent perturbations approaching packing-optimal codebooks.
Results
Reoptimization can yield a strictly smaller leading coefficient than every fixed packing-optimal codebook, while retaining the packing-optimal exponent.
Takeaways & Limitations
The simplex and orthoplex-bound ranges exhibit both equality and strict-improvement regimes for high-SNR error under SNR-wise codebook optimization.
Abstract
from arXiv · showhide
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 $K_{n+k,n}^\ast=4k(k-1)$.
I. INTRODUCTION
The paper asks whether reoptimizing spherical codebooks at every SNR can reduce the high-SNR error coefficient beyond what fixed packing-optimal codebooks achieve. It develops a soft-packing framework and demonstrates equality for simplex codes but strict improvement in part of the orthoplex-bound range.
- Fixed benchmark: For fixed packing-optimal codebooks, the leading coefficient equals the smallest ordered closest-pair count, equivalently the smallest average kissing number.This benchmark is obtained by holding one packing-optimal codebook fixed and minimizing its closest-pair count over the packing-optimal set.
- Motivation: Reoptimizing codeword locations at every SNR is distinct from optimizing among packing-optimal codebooks held fixed.Exponentially sensitive pairwise Gaussian errors make vanishing distance changes relevant to the leading coefficient.
- Main phenomenon: SNR-dependent families can retain the packing-optimal exponent while achieving a smaller leading coefficient than every fixed packing-optimal codebook.The improvement requires continued codebook motion with SNR and cannot be obtained by holding one codebook fixed.
- Orthoplex mechanism: In the orthoplex-bound construction, selected limiting closest pairs are made slightly closer while remaining pairs move farther apart on a larger vanishing scale.Selected pairs retain unit contributions, whereas the other pairs’ contributions vanish.
- Framework: The paper characterizes the SNR-wise optimum through a Gaussian soft-packing energy that uniformly represents normalized exact ML error.Near-minimizers within additive o(1) of the global soft-packing minimum are asymptotically ML-optimal.
B. Relation to Existing Theory
The paper extends fixed-codebook high-SNR theory from spherical packing to SNR-dependent codebook paths. It positions its soft-packing-to-exact-error transfer and pathwise leading-coefficient analysis as broader than earlier geometric and energy results.
- Scope: Classical fixed-codebook theory links high-SNR error to closest-pair geometry, while the paper studies codebooks reoptimized separately at each SNR.The unrestricted design varies the codebook with SNR rather than applying one fixed packing solution.
- Simplex theory: Earlier simplex results establish finite-SNR optimality of embedded regular simplices when n≥M−1, but do not address SNR-dependent paths outside the packing-optimal set.The paper uses simplex optimality as a concrete application rather than as its general framework.
- Riesz energies: Prior Riesz-energy work studies best-packing limits and exponent-dependent deformations, but does not transfer its asymptotics to exact AWGN ML error for general (n+k,n).The paper identifies the five-point square-pyramid case as a close antecedent to one specialization.
- Cross-entropy comparison: A positive-temperature cross-entropy result identifies a simplex-plus-antipodal-pairs geometry among packing-optimal orthoplex codes, but uses a different feasible class and objective.It does not analyze SNR-dependent paths outside the packing-optimal set.
- Paper contribution: The paper’s broader contribution is a uniform transfer from Gaussian soft-packing energy to exact ML error for SNR-dependent spherical-code designs.It also establishes existence, localization, and closest-pair contribution results.
B. Pairwise Geometry and the Packing Benchmark
For equal-energy spherical codebooks, maximizing pairwise correlation is equivalent to minimizing Euclidean distance, so spherical packing determines the best high-SNR exponent. The fixed benchmark then counts ordered closest pairs, while unrestricted SNR-wise optimization asks whether the coefficient itself can be reduced.
- Pairwise geometry: For unit-norm codewords, larger correlation means smaller Euclidean distance, making worst-correlation minimization equivalent to spherical packing.The optimal minimum squared distance is 2(1−ρ*_{M,n}).
- Fixed asymptotics: Packing-optimal codebooks attain the best high-SNR exponent, and their fixed-codebook leading coefficient is determined by ordered closest-pair multiplicity.Each unordered closest pair contributes both directed ordered indices.
- Fixed benchmark: K^fix_{M,n} is the minimum number of ordered closest-pair indices among packing-optimal codebooks, equivalently M times the minimum average kissing number.The same packing-optimal set supplies the exponent benchmark.
- Unrestricted design: The unrestricted finite-SNR optimum minimizes exact average ML error over the full compact manifold of labeled unit-vector codebooks.An optimizing codebook may change with SNR, unlike the fixed benchmark.
- Research question: The central question is whether the normalized unrestricted optimum has a finite limit and how that limit compares with the fixed packing-optimal coefficient.A strict inequality would mean equal exponents but a smaller exact error at sufficiently high SNR.
E. How K∗
The optimal SNR-dependent coefficient is governed by how a codebook family approaches a packing-optimal limit, not solely by the limiting packing’s closest-pair count. Gaussian soft-packing energy provides both a variational characterization and an asymptotically equivalent optimization objective.
- Pair contributions: A zero scaled offset gives a unit contribution, while divergence of the scaled offset to −∞ suppresses a pair’s contribution.Thus approach rates determine the coefficient assigned to each limiting closest pair.
- Orthoplex construction: Partitioning limiting closest pairs into two classes can preserve unit contributions for one class while making the other class’s contributions vanish.The first class is slightly closer, whereas the second is farther apart by a larger vanishing amount.
- Variational characterization: Minimizing over limiting packings and jointly feasible scaled offsets yields an exact variational characterization of K*_{M,n}.Near-minimizers of Gaussian soft-packing energy are asymptotically ML-optimal.
- Soft-packing control: A bounded Gaussian soft-packing energy forces the worst correlation into the 1/(nγ) scale, linking soft packing to the hard packing objective.This localization underpins the normalized exact-error analysis.
- Fixed paths: For a fixed packing-optimal codebook, every ordered closest pair contributes one, so its limiting soft-packing energy equals its ordered closest-pair count.This recovers the fixed-codebook benchmark K^fix_{M,n}.
B. Uniform Approximation of the Exact ML Error
Uniform soft-packing bounds transfer high-SNR behavior from Gaussian soft-packing energy to exact ML error, establishing existence and characterization of the optimal leading coefficient.
- Theorem 1 provides a uniform high-SNR transfer bound for Gaussian soft-packing energy over all codebooks with bounded energy.The constants depend on (M, n), L, and the packing separation, but not on the codebook or SNR.
- The optimized Gaussian soft-packing energy has a high-SNR limit that equals the optimized normalized exact ML error.Corollary 2 establishes asymptotically identical optimized normalized values for the two objectives.
- The initial finite-SNR equivalence alone does not establish convergence of the optimized value as SNR tends to infinity.The missing step is convergence of V_γ(M, n), supplied by the subsequent theorem.
- Theorem 2 proves that the optimal leading coefficient exists and equals the common limit of optimized soft-packing energy and normalized ML error.The globally optimal ML error obeys the corresponding leading-order expansion.
- Near-minimizers with Gaussian soft-packing energy within o(1) of the global minimum attain the optimal leading coefficient.Exact finite-SNR ML minimization is unnecessary for such an SNR-indexed family.
D. Comparison With the Best Packing-Optimal Benchmark
SNR-wise codebook optimization can exploit scaled geometric approaches to packing-optimal configurations, yielding leading coefficients below the fixed packing-optimal benchmark.
- The SNR-wise procedure preserves path-dependent pair contributions, whereas the fixed benchmark assigns one unit to every ordered closest pair.This distinction compares optimizing before the high-SNR limit with fixing a packing-optimal codebook first.
- Strict coefficient inequality makes the exact SNR-wise optimal error strictly smaller at every sufficiently large SNR.Equal coefficients instead make a fixed packing-optimal codebook leading-order optimal, though lower-order differences may remain.
- Any packing-tail-competitive family approaches the packing-optimal set, with maximal correlation within O((nγ)^−1) of the packing value.The family may still rotate, relabel codewords, or move among packing-optimal configurations.
- Only ordered pairs closest in the limiting optimal packing can contribute nonvanishing normalized error; non-closest pairs have exponentially vanishing contributions.The limiting packing identifies potentially relevant pairs, while finite-SNR displacement determines their weights.
- Different SNR-dependent approaches to the same packing can produce different limiting errors because nγ times the vanishing correlation offsets remains visible.Thus ordinary geometric convergence does not determine the leading coefficient by itself.
- A fixed packing-optimal codebook has coefficient equal to its ordered closest-pair count, while SNR-dependent offsets can assign fractional or zero pair contributions.For a constant family, every closest-pair offset is zero and contributes one.
C. Global Variational Characterization by Scaled Closest-Pair Offsets
The optimal leading coefficient is characterized variationally by jointly feasible scaled offsets of closest pairs across packing-optimal limiting codebooks.
- Proposition 2 minimizes weighted contributions over every packing-optimal limiting codebook and every jointly feasible scaled closest-pair-offset set.The minimum is attained, and the offsets cannot be optimized independently because they must arise from one codebook sequence.
- A constant packing has zero offsets and recovers the ordinary ordered closest-pair count as its leading coefficient.This embeds the fixed-codebook benchmark within the global variational characterization.
- The characterization shows that neither the packing nor its ordinary closest-pair count alone determines the optimal leading coefficient.Scaled approach data are additionally required.
- The variational characterization can be implicit because the admissible offset sets need not have tractable finite-dimensional descriptions.This is the stated scope boundary of the characterization.
- Matching lower and upper bounds identify a candidate value as the optimal coefficient and transfer it to the exact SNR-wise ML expansion.The construction must be a full SNR-indexed family whose soft-packing energy converges to the candidate.
VI. APPLICATION TO THE SIMPLEX CODES: EXACT OPTIMAL LEADING COEFFICIENTS
In the simplex range, regular simplices globally minimize Gaussian soft-packing energy and determine the exact optimal high-SNR leading coefficient without benefit from SNR-dependent motion.
- For 2 ≤ M ≤ n + 1, embedded regular (M − 1)-simplices are exactly the soft-packing-energy minimizers, up to orthogonal transformations and relabeling.The minimizer characterization holds for every positive SNR.
- No finite-SNR deformation, including an SNR-dependent one, can reduce the Gaussian soft-packing energy below the simplex value.The simplex is globally optimal for the soft-packing objective at every γ > 0.
- M(M − 1) is the optimal leading coefficient in the simplex range.The value is attained by the constant simplex family through matching bounds.
- Every ordered pair of distinct simplex vertices is a closest pair, determining the coefficient through the closest-pair count.Packing uniqueness identifies the regular simplex structure up to orthogonal transformation and relabeling.
- The optimal and best fixed-codebook leading coefficients coincide, so SNR-dependent motion gives no leading-order gain.A fixed regular simplex attains the optimum.
A. Packing Scale and Best Fixed-Codebook Leading Coefficient
The paper identifies the best fixed packing-optimal coefficient in the orthoplex-bound range and constructs SNR-dependent codebooks that can reduce the leading coefficient. At the terminal endpoint, the fixed full cross-polytope is optimal, while at the left endpoint the moving construction achieves a strict improvement.
- Best fixed-codebook coefficient: 4n(k −1) is the minimum ordered closest-pair count for fixed packing-optimal codebooks in the orthoplex-bound range.The minimum is attained by an orthogonal union of k −1 antipodal pairs and a regular simplex in the remaining subspace.
- SNR-dependent construction: The moving family suppresses contributions from one class of limiting closest pairs while retaining unit contributions from 4k(k −1) shifted-point pairs.The suppression uses two SNR-dependent rates: lowered correlations produce vanishing weights, whereas the remaining worsened correlations retain limiting weight one.
- Terminal endpoint k = n: 4n(n −1) is the terminal coefficient at k = n, where the fixed full cross-polytope is asymptotically ML-optimal and SNR-dependent motion cannot improve it.The full cross-polytope minimizes the Gaussian soft-packing energy at every positive SNR.
- Scope of the result: The exact finite-SNR result is established for the Gaussian soft-packing energy, whereas the exact ML conclusion is asymptotic.This distinction bounds the scope of the finite-SNR and ML claims.
- Left endpoint k = 2: 8 is the exact optimal leading coefficient at k = 2, attained by the fixed square for n = 2 and by the SNR-dependent construction for n ≥3.For n ≥3, the construction is the family specialized to k = 2.
D. Optimal Versus Best Fixed-Codebook Leading Coefficients
SNR-dependent codebooks can achieve a smaller leading coefficient than every fixed packing-optimal codebook while retaining the optimal exponent. The paper proves exact results for simplex and endpoint orthoplex-bound cases, gives an intermediate-range upper bound, and conjectures sharpness there.
- Orthoplex-bound range: At k = 2, a matching converse proves K* = 8 for n + 2 codewords, yielding a strict gain over the fixed benchmark when n ≥ 3.At the terminal endpoint k = n, the fixed cross-polytope attains the optimal coefficient and no leading-order gain occurs.
- Construction mechanism: The construction preserves the optimal exponent by making selected limiting closest pairs slightly closer while moving other pairs farther on a larger vanishing scale.Selected pairs contribute one each, whereas the improved pairs contribute vanishing amounts, allowing the moving family to outperform fixed packing-optimal codebooks.
- Orthoplex-bound range: For M = n + k with 2 ≤ k ≤ n, the best fixed-codebook coefficient is 4n(k − 1).This is the fixed benchmark for the orthoplex-bound range.
- Orthoplex-bound range: When 2 ≤ k < n, the explicit moving family has coefficient 4k(k − 1), exactly the fraction k/n of the best fixed-codebook coefficient.The SNR-wise optimum is no larger than the error of this constructed family.
- Orthoplex-bound range: For 3 ≤ k ≤ n − 1, the equality K* = 4k(k − 1) remains conjectural because no matching global lower bound has been proved.The explicit family establishes only the one-sided upper bound in this intermediate range.
APPENDIX A
Appendix A establishes the uniform transfer and compactness results needed to connect exact ML error with Gaussian soft-packing energy and characterize the optimal leading coefficient.
- The limiting energy characterizes K∗_{M,n} and matches the normalized SNR-wise minimum exact error.The two inequalities are established by selecting exact ML and soft-packing minimizers and applying uniform transfer.
- Theorem 1 supplies codebook-uniform asymptotics for exact ML error on bounded Gaussian soft-packing-energy classes.The proof combines Gaussian-tail estimates, pairwise tail sums, and Bonferroni bounds.
- Lemma 2 localizes any bounded-error SNR-indexed family to codebooks whose normalized packing gap satisfies nγξγ = O(1).This also ensures distinct codewords for sufficiently large γ.
- The optimized Gaussian soft-packing value Vγ(M,n) is eventually monotone, bounded, and convergent.Its lower bound is 2, while evaluating a packing-optimal codebook gives an upper bound NM.
E. Attainment by Gaussian Soft-Packing Energy Near-Minimization
Near-minimizing Gaussian soft-packing families converge toward packing-optimal codebooks, and their limiting closest-pair offsets determine the attainable leading coefficient.
- Any family attaining the optimal leading coefficient has accumulation points in the packing-optimal set.Compactness and continuity force the distance to the packing-optimal set to vanish.
- Only pairs closest in a limiting packing contribute to the limiting leading coefficient.Their contributions are determined by jointly feasible scaled correlation offsets.
- Simplex range: For 2 ≤ M ≤ n + 1, the embedded regular simplex uniquely minimizes the Gaussian soft-packing energy and yields coefficient M(M−1).The equality case forces zero centroid and equal off-diagonal correlations.
- Orthoplex-bound range: For M = n + k with 2 ≤ k ≤ n, every packing-optimal codebook has at least the stated ordered closest-pair count, while a simplex-plus-antipodal-pairs construction attains 4n(k−1).The construction uses a regular simplex and k−1 mutually orthogonal antipodal pairs.
C. Multiscale Construction in the Nonterminal Orthoplex-Bound Range
The nonterminal orthoplex construction uses multiscale perturbations to retain the optimal exponent while concentrating the leading coefficient on one pair class.
- The construction converges to a deleted cross-polytope while remaining a valid spherical codebook of size n + k.Positive definiteness and distinctness hold for sufficiently large γ.
- The perturbed codebook has maximum correlation pγ, attained by 4k(k−1) ordered cross-axis pairs.For sufficiently large γ, pγ > 0 > −qγ and 2pγ − 1 < 0.
- The remaining pair classes have correlations whose contributions vanish in the high-SNR limit.Thus the constructed family’s leading coefficient converges to 4k(k−1).
- Consequently, K∗_{n+k,n} ≤ 4k(k−1) for 2 ≤ k < n.The family has uniformly bounded Gaussian soft-packing energy, enabling transfer to exact ML error.