Improved Linear Programming Bounds for Antipodal Spherical Codes

Let S ?1; 1). A nite set C = fx i g M i=1 < n is called a spherical S-code if kx i k = 1 for each i, and x T i x j 2 S, i 6 = j. For S = ?1; :5] maximizing M = jCj 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 modiication of the bounding procedure that is applicable to antipodal codes; that is, codes where x 2 C) ?x 2 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 n .