Uniacute Spherical Codes

Abstract

A spherical L-code, where \(L \subseteq [-1,\infty )\), consists of unit vectors in \(\mathbb {R}^d\) whose pairwise inner products are contained in L. Determining the maximum cardinality \(N_L(d)\) of an L-code in \(\mathbb {R}^d\) 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 \(L = [-1, -\beta ] \cup \{\alpha \}\) with \(\alpha ,\beta > 0\) (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 \(\limsup _{d\rightarrow \infty } N_L(d)/d \le 2p\) for \(L = [-1, -\beta ] \cup \{\alpha \}\) and \(p = \lfloor \alpha /\beta \rfloor + 1\). 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.