New Bounds for Spherical Two-Distance Sets

Abstract

A spherical two-distance set is a finite collection of unit vectors in such that the distance between two distinct vectors assumes one of only two values. We use the semidefinite programming method to compute improved estimates of the maximum size of spherical two-distance sets. Exact answers are found for dimensions n=23 and 40⩽n⩽93 (n≠46, 78), where previous results gave divergent bounds.

Keywords:

• two-distance sets
• positive definite matrices
• semidefinite programming

2000 AMS Subject Classification:

• 52C35
• 94B75

Acknowledgments

We are grateful to Chao-Wei Chen and Johan Löfberg for their significant help with the Matlab implementation. Research supported in part by NSF grants DMS1101697, CCF0916919, and CCF1217894, and NSA grant H78230-12-1-0260.

Notes

¹ SOSTOOLS is available at http://www.cds.caltech.edu/sostools/ , YALMIP at http://users.isy.liu.se/johanl/yalmip/ .