Synopses & Reviews
The third edition of this definitive and popular book continues to pursue the question: what is the most efficient way to pack a large number of equal spheres in n-dimensional Euclidean space? The authors also examine such related issues as the kissing number problem, the covering problem, the quantizing problem, and the classification of lattices and quadratic forms. There is also a description of the applications of these questions to other areas of mathematics and science such as number theory, coding theory, group theory, analogue-to-digital conversion and data compression, n-dimensional crystallography, dual theory and superstring theory in physics. New and of special interest is a report on some recent developments in the field, and an updated and enlarged supplementary bibliography with over 800 items.
Review
Third Edition J.H. Conway and N.J.A. Sloane Sphere Packings, Lattices and Groups "This is the third edition of this reference work in the literature on sphere packings and related subjects. In addition to the content of the preceding editions, the present edition provides in its preface a detailed survey on recent developments in the field, and an exhaustive supplementary bibliography for 1988-1998. A few chapters in the main text have also been revised."--MATHEMATICAL REVIEWS
Review
Third Edition
J.H. Conway and N.J.A. Sloane
Sphere Packings, Lattices and Groups
"This is the third edition of this reference work in the literature on sphere packings and related subjects. In addition to the content of the preceding editions, the present edition provides in its preface a detailed survey on recent developments in the field, and an exhaustive supplementary bibliography for 1988-1998. A few chapters in the main text have also been revised."--MATHEMATICAL REVIEWS
Synopsis
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.
Synopsis
This third edition of a definitive and popular book continues to pursue the question "What is the most efficient way to pack a large number of equal spheres in n-dimensional Euclidean space?" as well as related areas such as the kissing number problem.
Table of Contents
Preface to First Edition.- Preface to Third Edition.- List of Symbols.- Sphere Packings and Kissing Numbers.- Coverings, Lattices and Quantizers.- Codes, Designs, and Groups.- Certain Important Lattices and Their Properties.- Sphere Pakcking and Error-Correcting Codes.- Laminated Lattices.- Further Connections Between Codes and Lattices.- Algebraic Constructions for Lattices.- Bounds for Codes and Sphere Packings.- Three Lectures on Exceptional Groups.- The Golay Codes and the Mathieu Groups.- A Characterization of the Leech Lattice.- Bounds on Kissing Numbers.- Uniqueness of Certain Spherical Codes.- On the Classification of Integral Quadratic Forms.- Enumeration of Unimodular Lattices.- The 24-Dimensional Odd Unimodular Lattices.- Even Unimodular 24-Dimensional Lattices.- Enumeration of Extremal Self-Dual Lattices.- Finding the Closest Lattice Point.- Voronoi Cells of Lattices and Quantization Errors.- A Bound for the Covering Radius of the Leech Lattice. The Covering Radius of the Leech Lattice.- Twenty-Three Constructions for the Leech Lattice.- The Cellular Structure of the Leech Lattice.- Lorenzian Forms for the Leech Lattice.- The Automorphism Group of the 26-Dimensional Even Unimodular Lorenzian Lattice.- Leech Roots and Vinberg Groups.- The Moster Group and its 196885-Dimensional Space.- A Monster Lie Algebra? Bibliography. Supplemental Bibliography.