Improved Linear Programming Bounds for Antipodal Spherical Codes

   Abstract. Let Ssubset[-1,1) . A finite set Ccal=set x _{i i=1} ^M subsetRe ⁿ is called a spherical S-code if norm x _i =1 for each i , and x _i tran x _j ∈ S , ine j . For S=[-1, 0.5] maximizing M=|C| 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∈Ccalimplies -x∈Ccal . 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 Lam _n .