Lifting to Higher Dimensions: How OpenAI Shattered 1978 Limits on Sphere Packing and Error-Correcting Codes

In geometric optimization, few problems possess the elegance and difficulty of dense sphere packing in Euclidean space (\mathbb{R}^d). In 1978, Kabatianskii and Levenshtein established the asymptotic upper bound on packing density. For nearly half a century, that exponent resisted all analytical assaults.

1. The Cohn-Elkies Linear Programming Limit

In 2003, Henry Cohn and Noam Elkies revolutionized the field by showing that radial Schwartz functions with non-positive Fourier transforms bound the sphere packing density. The first chapter of OpenAI's anthology establishes the exact asymptotic value of this program:

This achieves an asymptotic packing exponent of approximately **0.6044**, beating the 1978 Kabatianskii-Levenshtein exponent of **0.59905576**. More fundamentally, the proof proves that no auxiliary function within the Cohn-Elkies framework can ever surpass this limit.

2. The Moving-Subspace Projection Method

In classical Delsarte linear programming for binary codes, a single one-dimensional line is attached to each codeword. The second chapter replaces this static line with an exponentially high-dimensional subspace that moves equivariantly with the codeword (x).

By constructing orthogonal projections (P_x) such that the trace overlap (\operatorname{tr}(P_x P_y)) remains a scalar function of Hamming distance, the positive-definite matrix structure is preserved while multiplying coding-rate bounds by exponential factors.