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" Sphere Packings, Lattices and Groups "


Document Type : BL
Record Number : 621229
Doc. No : dltt
Main Entry : Conway, J. H.
Title & Author : Sphere Packings, Lattices and Groups\ by J. H. Conway, N. J. A. Sloane.
Edition Statement : Third Edition.
Publication Statement : New York, NY :: Springer New York :: Imprint: Springer,, 1999.
Series Statement : Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics,; 290
ISBN : 9781475765687
: : 9781441931344
Contents : 1 Sphere Packings and Kissing Numbers -- 2 Coverings, Lattices and Quantizers -- 3 Codes, Designs and Groups -- 4 Certain Important Lattices and Their Properties -- 5 Sphere Packing and Error-Correcting Codes -- 6 Laminated Lattices -- 7 Further Connections Between Codes and Lattices -- 8 Algebraic Constructions for Lattices -- 9 Bounds for Codes and Sphere Packings -- 10 Three Lectures on Exceptional Groups -- 11 The Golay Codes and the Mathieu Groups -- 12 A Characterization of the Leech Lattice -- 13 Bounds on Kissing Numbers -- 14 Uniqueness of Certain Spherical Codes -- 15 On the Classification of Integral Quadratic Forms -- 16 Enumeration of Unimodular Lattices -- 17 The 24-Dimensional Odd Unimodular Lattices -- 18 Even Unimodular 24-Dimensional Lattices -- 19 Enumeration of Extremal Self-Dual Lattices -- 20 Finding the Closest Lattice Point -- 21 Voronoi Cells of Lattices and Quantization Errors -- 22 A Bound for the Covering Radius of the Leech Lattice -- 23 The Covering Radius of the Leech Lattice -- 24 Twenty-Three Constructions for the Leech Lattice -- 25 The Cellular Structure of the Leech Lattice -- 26 Lorentzian Forms for the Leech Lattice -- 27 The Automorphism Group of the 26-Dimensional Even Unimodular Lorentzian Lattice -- 28 Leech Roots and Vinberg Groups -- 29 The Monster Group and its 196884-Dimensional Space -- 30 A Monster Lie Algebra? -- Supplementary Bibliography.
Abstract : We now apply the algorithm above to find the 121 orbits of norm -2 vectors from the (known) nann 0 vectors, and then apply it again to find the 665 orbits of nann -4 vectors from the vectors of nann 0 and -2. The neighbors of a strictly 24 dimensional odd unimodular lattice can be found as follows. If a norm -4 vector v E II . corresponds to the sum 25 1 of a strictly 24 dimensional odd unimodular lattice A and a !-dimensional lattice, then there are exactly two nonn-0 vectors of ll25,1 having inner product -2 with v, and these nann 0 vectors correspond to the two even neighbors of A. The enumeration of the odd 24-dimensional lattices. Figure 17.1 shows the neighborhood graph for the Niemeier lattices, which has a node for each Niemeier lattice. If A and B are neighboring Niemeier lattices, there are three integral lattices containing A n B, namely A, B, and an odd unimodular lattice C (cf. [Kne4]). An edge is drawn between nodes A and B in Fig. 17.1 for each strictly 24-dimensional unimodular lattice arising in this way. Thus there is a one-to-one correspondence between the strictly 24-dimensional odd unimodular lattices and the edges of our neighborhood graph. The 156 lattices are shown in Table 17 .I. Figure I 7. I also shows the corresponding graphs for dimensions 8 and 16.
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