Asymptotic rate bounds for spherical codes
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How large can a spherical code be at a fixed angular separation, and a binary code at fixed relative Hamming distance? The asymptotic rate is the exponential growth rate as dimension or block length increases.
Why it matters and what counts as progress
Why it matters: Sharper upper bounds constrain code size beyond the classical linear-programming benchmarks.
What counts as progress: A proved rate bound improving the corresponding MRRW or spherical localized linear-programming bound across a stated parameter interval.
How the problem developed
- 1978 · Upper bound [1]
Grigory Kabatiansky and Vladimir Levenshtein Used zonal spherical functions to obtain a stronger asymptotic upper bound on spherical-code size for each fixed angular separation.
OpenAI Astra Reported improved high-dimensional spherical-code bounds through a localized linear-programming hierarchy for every fixed parameter 0<s<1.
AI contribution
Ten Advances in Mathematics and Theoretical Computer Science
For every relative binary distance 0<δ<1/2, Chapter 2 gives an asymptotic rate bound strictly below the classical MRRW bound. Companion spherical-code bounds improve the localized linear-programming hierarchy for 0<s<1.
Sources
- Kabatiansky and Levenshtein, On Bounds for Packings on a Sphere and in Space
- OpenAI, Ten Advances in Mathematics, Chapter 2
- OpenAI, Ten Advances in Mathematics release and attribution
- OpenAI, code-bound formalization manifest
- OpenAI, binary-code comparator statement
- OpenAI, spherical-code comparator statement
- Sienicki and coauthors, Chapter 2 human review
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