Bounds for spherical codes

TL;DR

This paper proves a conjecture regarding the maximum size of spherical codes with fixed angles, establishing an upper bound of O(d^k) lines in high-dimensional space for any fixed set of angles.

Contribution

It confirms Bukh's conjecture by providing an explicit bound on the size of spherical codes with fixed angles in Euclidean space.

Findings

Maximum size of spherical codes is O(d^k) for fixed angles.

Established bounds for spherical codes with prescribed inner products.

Extended understanding of geometric configurations in high dimensions.

Abstract

A set $C$ of unit vectors in $\mathbb{R}^d$ is called an $L$-spherical code if $x \cdot y \in L$ for any distinct $x,y$ in $C$. Spherical codes have been extensively studied since their introduction in the 1970's by Delsarte, Goethals and Seidel. In this note we prove a conjecture of Bukh on the maximum size of spherical codes. In particular, we show that for any set of $k$ fixed angles, one can choose at most $O(d^k)$ lines in $\mathbb{R}^d$ such that any pair of them forms one of these angles.