Sphere Packings, Lattices and Groups (Grundlehren der mathematischen Wissenschaften, 290)

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. The Third Edition Of This Timely, 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 Continue To Examine Related Problems Such As The Kissing Number Problem, The Covering Problem, The Quantizing Problem, And The Classification Of Lattices And Quadratic Forms. Like The Previous Edition, The Third Edition Describes The Applications Of These Questions To Other Areas Of Mathematics And Science Such As Number Theory, Coding Theory, Group Theory, Analog-to-digital Conversion And Data Compression, N-dimensional Crystallography, Dual Theory And Superstring Theory In Physics. Of Special Interest To The Third Edtion Is A Brief Report On Some Recent Developments In The Field And An Updated And Enlarged Supplementary Bibliography With Over 800 Items. 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. By J. H. Conway, N. J. A. Sloane. Front Matter....Pages i-lxxiv Sphere Packings and Kissing Numbers....Pages 1-30 Coverings, Lattices and Quantizers....Pages 31-62 Codes, Designs and Groups....Pages 63-93 Certain Important Lattices and Their Properties....Pages 94-135 Sphere Packing and Error-Correcting Codes....Pages 136-156 Laminated Lattices....Pages 157-180 Further Connections Between Codes and Lattices....Pages 181-205 Algebraic Constructions for Lattices....Pages 206-244 Bounds for Codes and Sphere Packings....Pages 245-266 Three Lectures on Exceptional Groups....Pages 267-298 The Golay Codes and The Mathieu Groups....Pages 299-330 A Characterization of the Leech Lattice....Pages 331-336 Bounds on Kissing Numbers....Pages 337-339 Uniqueness of Certain Spherical Codes....Pages 340-351 On the Classification of Integral Quadratic Forms....Pages 352-405 Enumeration of Unimodular Lattices....Pages 406-420 The 24-Dimensional Odd Unimodular Lattices....Pages 421-428 Even Unimodular 24-Dimensional Lattices....Pages 429-440 Enumeration of Extremal Self-Dual Lattices....Pages 441-444 Finding the Closest Lattice Point....Pages 445-450 Voronoi Cells of Lattices and Quantization Errors....Pages 451-477 A Bound for the Covering Radius of the Leech Lattice....Pages 478-479 The Covering Radius of the Leech Lattice....Pages 480-507 Twenty-Three Constructions for the Leech Lattice....Pages 508-514 The Cellular Structure of the Leech Lattice....Pages 515-523 Lorentzian Forms for the Leech Lattice....Pages 524-528 The Automorphism Group of the 26-Dimensional Lorentzian Lattice....Pages 529-533 Leech Roots and Vinberg Groups....Pages 534-555 The Monster Group and its 196884-Dimensional Space....Pages 556-569 A Monster Lie Algebra?....Pages 570-573 Back Matter....Pages 574-706 This book is an exposition of the mathematics arising from the theory of sphere packings. Considerable progress has been made on the basic problems in the field, and the most recent research is presented here. Connections with many areas of pure and applied mathematics, for example signal processing, coding theory, are thoroughly discussed.