, Volume 28, Issue 1, pp 107-114
Date: 01 Jul 2002

Improved Linear Programming Bounds for Antipodal Spherical Codes

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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 ∥x i ∥ =1 for each i, and x i T x j S, 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 Λ n .