Discrete & Computational Geometry

, Volume 28, Issue 1, pp 107–114

Improved Linear Programming Bounds for Antipodal Spherical Codes

  • Anstreicher
Article

DOI: 10.1007/s00454-001-0080-5

Cite this article as:
Anstreicher Discrete Comput Geom (2002) 28: 107. doi:10.1007/s00454-001-0080-5

Abstract

Let S ⊂ [-1,1). A finite set \(\mathcal{C} = \{ x^i \} _{i = 1}^M \subset \Re ^n\) is called a spherical S-code if ∥xi∥ =1 for each i, and xiTxjS, ij. For S = [−1, 0.5] maximizing \(M = \left| \mathcal{C} \right|\) is commonly referred to as the kissing number problem. A well-known technique based on harmonic analysis and linear programming can be used to bound M. We consider a modification of the bounding procedure that is applicable to antipodal codes; that is, codes where \(x \in \mathcal{C} \Rightarrow - x \in \mathcal{C}\). Such codes correspond to packings of lines in the unit sphere, and include all codes obtained as the collection of minimal vectors in a lattice. We obtain improvements in upper bounds for kissing numbers attainable by antipodal codes in dimensions 16 ≤ n ≤ 23. We also show that for n = 4, 6 and 7 the antipodal codes with maximal kissing numbers are essentially unique, and correspond to the minimal vectors in the laminated lattices Λn.

Copyright information

© Springer-Verlag New York Inc. 2002

Authors and Affiliations

  • Anstreicher
    • 1
  1. 1.Department of Management Sciences, University of Iowa, Iowa City, IA 52242, USA kurt-anstreicher@uiowa.eduUSA