Bounds on equiangular lines and on related spherical codes

TL;DR

This paper establishes linear bounds on the size of certain spherical codes and equiangular line sets, advancing understanding of their maximal configurations in high-dimensional spaces.

Contribution

It provides new linear bounds for $L$-spherical codes with specific inner product sets, including equiangular lines, which were previously less understood.

Findings

Linear bounds for $L$-spherical codes with fixed inner products

Bounds apply to equiangular lines with fixed angles

Results improve understanding of maximal configurations in high dimensions

Abstract

An $L$-spherical code is a set of Euclidean unit vectors whose pairwise inner products belong to the set $L$. We show, for a fixed $\alpha,\beta>0$, that the size of any $[-1,-\beta]\cup\{\alpha\}$-spherical code is at most linear in the dimension. In particular, this bound applies to sets of lines such that every two are at a fixed angle to each another.