Spherical two-distance sets

Details Spherical two-distance sets Spherical two-distance sets

2008-01-24

arxiv.org/abs/0801.3706v3

Journal of Combinatorial Theory, Series A 116 (2009) 988--995

Mathematics

Metric Geometry

9 pages, (v2) several small changes and corrections suggested by referees, accepted in Journal of Combinatorial Theory, Series

Scientific paper

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

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