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.
Inhaltsverzeichnis
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.
