Uniacute Spherical Codes

A spherical L-code, where L subset of[-1,infinity), consists of unit vectors in R-d whose pairwise inner products are contained in L. Determining the maximum cardinality N-L(d) of an L-code in R-d 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={-alpha, alpha}, is a rare and notable solved case. Bukh studied an extension of equiangular lines and showed that N-L(d)=O-L(d) for L=[-1,-beta]boolean OR{alpha} with alpha, beta>0 (we call such L-codes "uniacute"), leaving open the question of determining the leading constant factor. Balla, Draxler, Keevash, and Sudakov proved a "uniform bound" showing lim sup(d ->infinity )N(L)(d)/d <= 2p for L=[-1,-beta]boolean OR{alpha} and p=& LeftFloor;alpha/beta & RightFloor;+1. For which (alpha, beta) 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.