, Volume 28, Issue 1, pp 107-114

Improved Linear Programming Bounds for Antipodal Spherical Codes

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Abstract. Let S\subset[-1,1) . A finite set \Ccal=\set x i i=1 M \subset\Re n is called a spherical S-code if \norm x i =1 for each i , and x i \tran x j ∈ S , i\ne 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∈\Ccal\implies -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 .