Spherical two-distance sets

TL;DR

This paper investigates the maximum size of spherical two-distance sets in Euclidean space, establishing new bounds and exact values for certain dimensions using polynomial and Delsarte's methods.

Contribution

It proves new upper bounds for the size of spherical two-distance sets when the sum of the inner products is nonnegative and refines known bounds for specific dimensions.

Findings

g(n)=L(n) for 6<n<22 and 23<n<40

g(23) is either 276 or 277

Upper bounds are established for cases where a+b<0

Abstract

A set S of unit vectors in n-dimensional Euclidean space is called spherical two-distance set, if there are two numbers a and b, and inner products of distinct vectors of S are either a or b. The largest cardinality g(n) of spherical two-distance sets is not exceed n(n+3)/2. This upper bound is known to be tight for n=2,6,22. The set of mid-points of the edges of a regular simplex gives the lower bound L(n)=n(n+1)/2 for g(n. In this paper using the so-called polynomial method it is proved that for nonnegative a+b the largest cardinality of S is not greater than L(n). For the case a+b<0 we propose upper bounds on |S| which are based on Delsarte's method. Using this we show that g(n)=L(n) for 6<n<22, 23<n<40, and g(23)=276 or 277.