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 ${\mathrm{limsup}}_{d\to \infty }{N}_L(d)/d\le 2p$ 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.

26 pages, 8 figures