By Paula Ribenboim

ISBN-10: 0471718041

ISBN-13: 9780471718048

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**Example text**

42], [Wel4, p. 731. 7937 ... 5612... and R = respectively. On the other hand the Voronoi cell for the bcc lattice is a truncated octahedron (Fig. 2b) , one of the Archimedean polyhedra [Cunl, p. 981, [Fej91, [Holll, [Loel, p. 1291, [WeI4, p. 731, [Wenl, p. 211. 5456 ... 7043 .... Thus although the fcc lattice is the better packing, the bcc lattice is indeed a better covering. There is another difference between these two lattices. In the bcc lattice, as in the planar hexagonal lattice, there is only one kind of hole (all holes are deep), but in the fcc lattice there are two kinds (shallow and deep holes).

1. Covering the plane with circles. In (a) the centers belong to the square lattice Z2, in (b) they belong to the hexagonal lattice. (b) is a more efficient or thinner covering. 2 The covering radius and the Voronoi cells. Consider any discrete collection of points [lj> = {p b P 2 , . } in Rn. The least upper bound for the distance from any point of Rn to the closest point Pi is called the covering radius of [lj>, usually denoted by R. Thus R = sup inf dist(x ,p). ) Then spheres of radius R centered at the points of [lj> will cover Rn, and no smaller radius will do.

Any other spherical code obtained by the construction is a subset of this. The Golay code C(J24 (Chap. S29°. Thus A (24,coS-11/3) ~ 4096. By using certain codes constructed by Justesen, Weldon, and Sugiyama et al. (Chap. 6), we may obtain infinite sequences of spherical codes, with n --+ 00, having a fixed minimal angle IjJ and containing 2cn points (where c depends only on 1jJ). 110 ... 73". For larger angles the codes of Weldon and Sugiyama et al. must be used. The kissing number problem for example requires IjJ = 60 0, and from the Sugiyama et al.

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