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, Dr\"{a}xler, Keevash, and Sudakov proved a "uniform bound" showing $\limsup_{d\to\infty} N_L(d)/d \le 2p$ 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.