Uniacute Spherical Codes
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Author(s)
Lepsveridze, Saba
Saatashvili, Aleksandre
Zhao, Yufei
Date Issued
January 2, 2025
Journal
Combinatorica
Publisher
Springer Berlin Heidelberg
Citation
Lepsveridze, S., Saatashvili, A. & Zhao, Y. Uniacute Spherical Codes. Combinatorica 45, 8 (2025).
Version
Author's final manuscript
Abstract
A spherical L-code, where L ⊆ [−1,∞), consists of unit vectors in Rd whose pairwise inner products are contained in L. Determining the maximum cardinality NL (d)
of an L-code in Rd is a fundamental question in discrete geometry and has been
extensively investigated for various choices of L. Our understanding in high dimensions is generally quite poor. Equiangular lines, corresponding to L = {−α, α}, is
a rare and notable solved case. Bukh studied an extension of equiangular lines and
showed that NL (d) = OL (d) for L = [−1, −β]∪{α} with α, β > 0 (we call such
L-codes “uniacute”), leaving open the question of determining the leading constant
factor. Balla, Dräxler, Keevash, and Sudakov proved a “uniform bound” showing
lim supd→∞ NL (d)/d ≤ 2p for L = [−1, −β]∪{α} and p = α/β + 1. For which
(α, β) is this uniform bound tight? We completely answer this question. We develop a
framework for studying uniacute codes, including a global structure theorem showing
that the Gram matrix has an approximate p-block structure. We also formulate a notion
of “modular codes,” which we conjecture to be optimal in high dimensions.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00493-024-00125-z
