2023/11/29 by Lepsveridze, Saba, Saatashvili, Aleksandre, Zhao, Yufei
#Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2311.17734
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 \limsupd→∞ NL(d)/d ≤ 2p for L = [-1, -β] ∪ \α\ and p = \lfloor α/β\rfloor + 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.