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Packings with large minimum kissing numbers. (English) Zbl 0894.52008

The minimum kissing number of a collection of objects in \(R^n\) with pairwise disjoint interiors is the minimum, over all objects \(B\) in the collection, of the number of other objects touched by \(B\). It is known that a lattice packing of unit balls in \(R^n\) may have a minimum kissing number of at least \(2^{\Omega (\log^2n)}\). Here, the author constructs, where \(n=4^k\), a finite collection of balls with a kissing number greater than \(2^{\sqrt n}\). It may be seen that this is a very significant increase over the best known result for a lattice packing.
The first step is to construct, for each \(k\geq 2\), a linear, binary error correcting code of length \(4^k\), dimension \(k(2^{k-1} +1)\) and minimum distance \(4^{k-1}\), in which the number of words of minimum Hamming weight is greater than \(2^{\sqrt n}\). The desired construction is given by a set of balls, centered at the vectors of the code, and of radius \(2^{k-2}\).
The author conjectures, in closing, that further order-of-magnitude improvements remain to be made; in particular, that there exist lattice packings in \(R^n\) with kissing number \(2^{\Omega (n)}\).
Reviewer: R.Dawson (Halifax)

MSC:

52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
94B05 Linear codes (general theory)
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References:

[1] Alon, N.; Goldreich, O.; Hastad, J.; Peralta, R., Random Structures and Algorithms, 3, 289-304 (1992) · Zbl 0755.60002
[2] Barnes, E. S.; Sloane, N. J.A., New lattice packings of spheres, Canad. J. Math., 35, 30-41 (1983) · Zbl 0509.52010
[3] Barnes, E. S.; Wall, G. E., Some extreme forms defined in terms of Abelian groups, J. Austral. Math. Soc., 1, 47-63 (1959) · Zbl 0109.03304
[4] Bos, A.; Conway, J. H.; Sloane, N. J.A., Further lattice packings in high dimensions, Mathematika, 29, 171-180 (1982) · Zbl 0489.52017
[5] Conway, J. H.; Sloane, N. J.A., Sphere Packings, Lattices and Groups (1988), Springer: Springer Berlin · Zbl 0634.52002
[6] Leech, J., Some sphere packings in higher space, Canad. J. Math., 16, 657-682 (1984) · Zbl 0142.20201
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