Uniacute Spherical Codes

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.