Math Conjectures

Asymptotic Spherical-Code Rate

OPENMajorOpen problemProposed 1959 · Standard version

Canonical statement

For −1<s<1-1<s<1, let AS(d,s)A_S(d,s) be the largest number of unit vectors in Rd\mathbb R^d whose pairwise inner products are at most ss. Determine the asymptotic rate
RS(s)=lim sup⁡d→∞1dlog⁡2AS(d,s) R_S(s)=\limsup_{d\to\infty}\frac1d\log_2 A_S(d,s)
for every fixed ss, and determine where the corresponding limit exists.
View source LaTeX
For \(-1<s<1\), let \(A_S(d,s)\) be the largest number of unit vectors in \(\mathbb R^d\) whose pairwise inner products are at most \(s\). Determine the asymptotic rate
\[
  R_S(s)=\limsup_{d\to\infty}\frac1d\log_2 A_S(d,s)
\]
for every fixed \(s\), and determine where the corresponding limit exists.
INFO-005:[Shannon1959SpherePacking][KabatianskyLevenshtein1978Bounds][OpenAI2026TenAdvances]
Information theoryDiscrete geometry

Notes

Spherical codes model finite signal sets with bounded pairwise correlation. Shannon introduced their asymptotic rate in connection with Gaussian channels [Shannon1959SpherePacking], and Kabatiansky–Levenshtein linear programming supplies the classical high-dimensional upper regime [KabatianskyLevenshtein1978Bounds]. The August 1, 2026 manuscript claims exponential upper-bound improvements [OpenAI2026TenAdvances], but the exact rate function remains far from determined.

Proof-claim watch (1)

Exponential improvements for binary and spherical code upper bounds

Grade C

Substantive progress claim pending verification; it does not settle either exact rate function or beat the binary Gilbert–Varshamov lower curve.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.