Sphere Packings, Lattices and Groups

Transcription

1 J.H. Conway N.J.A. Sloane Sphere Packings, Lattices and Groups Third Edition With Additional Contributions by E. Bannai, R.E. Borcherds, J. Leech, S.P. Norton, A.M. Odlyzko, R.A. Parker, L. Queen and B.B. Venkov With 112 Illustrations Springer

2 Contents Preface to First Edition Preface to Third Edition List of Symbols v xv lxi Chapter 1 Sphere Packings and Kissing Numbers J.H. Conway and N.J.A. Sloane 1 The Sphere Packing Problem I I Packing Ball Bearings 1.2 Lattice Packings 3 3 Nonlattice Packings 7 4 /7-Dimensional Packings 8 5 Sphere Packing Problem Summary of Results The Kissing Number Problem The Problem of the Thirteen Spheres Kissing Numbers in Other Dimensions Spherical Codes The Construction of Spherical Codes from Sphere Packings The Construction of Spherical Codes from Binary Codes Bounds on A(/i,«j>) 27 Appendix: Planetary Perturbations 29 Chapter 2 Coverings, Lattices and Quantizers J.H. Conway and N.J.A. Sloane The Covering Problem Covering Space with Overlapping Spheres The Covering Radius and the Voronoi Cells Covering Problem Summary of Results Computational Difficulties in Packings and Coverings 40

3 lxv Contents 2. Lattices, Quadratic Forms and Number Theory The Norm of a Vector Quadratic Forms Associated with a Lattice Theta Series and Connections with Number Theory Integral Lattices and Quadratic Forms Modular Forms Complex and Quaternionic Lattices Quantizers Quantization, Analog-to-Digital Conversion and Data Compression The Quantizer Problem Quantizer Problem Summary of Results 59 Chapter 3 Codes, Designs and Groups J.H. Conway and N.J.A. Sloane The Channel Coding Problem The Sampling Theorem Shannon's Theorem Error Probability Lattice Codes for the Gaussian Channel Error-Correcting Codes The Error-Correcting Code Problem Further Definitions from Coding Theory Repetition, Even Weight and Other Simple Codes Cyclic Codes BCH and Reed-Solomon Codes Justesen Codes Reed-Muller Codes Quadratic Residue Codes Perfect Codes The Pless Double Circulant Codes Goppa Codes and Codes from Algebraic Curves Nonlinear Codes Hadamard Matrices r-designs, Steiner Systems and Spherical f-designs /-Designs and Steiner Systems Spherical r-designs The Connections with Group Theory The Automorphism Group of a Lattice Constructing Lattices and Codes from Groups 92 Chapter 4 Certain Important Lattices and Their Properties J.H. Conway and N.J.A. Sloane Introduction Reflection Groups and Root Lattices Gluing Theory 99

4 Contents lxvi 4. Notation; Theta Functions Jacobi Theta Functions The ^-Dimensional Cubic Lattice Z" The ^-Dimensional Lattices A a and A* The Lattice A n The Hexagonal Lattice The Face-Centered Cubic Lattice The Tetrahedral or Diamond Packing The Hexagonal Close-Packing The. Dual Lattice A* The Body-Centered Cubic Lattice The H-Dimensional Lattices > and D* The Lattice D n The Four-Dimensional Lattice D The Packing D n The Dual Lattice D* The Lattices E b, E 7 and The 8-Dimensional Lattice The 7-Dimensional Lattices 7 and E? The 6-Dimensional Lattices 6 and Et The 12-Dimensional Coxeter-Todd Lattice K n The 16-Dimensional Barnes-Wall Lattice A, The 24-Dimensional Leech Lattice A Chapter 5 Sphere Packing and Error-Correcting Codes J. Leech and N.J.A. Sloane Introduction The Coordinate Array of a Point Construction A The Construction Center Density Kissing Numbers Dimensions 3 to Dimensions 7 and Dimensions 9 to Comparison of Lattice and Nonlattice Packings Construction B The Construction Center Density and Kissing Numbers Dimensions 8, 9 and Dimensions 15 to Packings Built Up by Layers Packing by Layers Dimensions 4 to Dimensions 11 and 13 to Density Doubling and the Leech Lattice A Cross Sections of A Other Constructions from Codes 146

5 lxvii Contents 5.1 A Code of Length A Lattice Packing in R Cross Sections of A Packings Based on Ternary Codes Packings Obtained from the Pless Codes Packings Obtained from Quadratic Residue Codes Density Doubling in R 24 and R Construction C The Construction Distance Between Centers Center Density Kissing Numbers Packings Obtained from Reed-Muller Codes Packings Obtained from BCH and Other Codes Density of BCH Packings Packings Obtained from Justesen Codes 155 Chapter 6 Laminated Lattices J.H. Conway and N.J.A. Sloane Introduction The Main Results Properties of A o to A Dimensions 9 to The Deep Holes in A, Dimensions 17 to Dimensions 25 to Appendix: The Best Integral Lattices Known 179 Chapter 7 Further Connections Between Codes and Lattices N.J.A. Sloane Introduction Construction A Self-Dual (or Type I) Codes and Lattices.: Extremal Type I Codes and Lattices ConstructionB Type II Codes and Lattices Extremal Type II Codes and Lattices Constructions A and B for Complex Lattices Self-Dual Nonbinary Codes and Complex Lattices Extremal Nonbinary Codes and Complex Lattices 205 Chapter 8 Algebraic Constructions for Lattices J.H. Conway and N.J.A. Sloane Introduction The Icosians and the Leech Lattice 207

6 Contents lxviii 2.1 The Icosian Group The Icosian and Turyn-Type Constructions for the Leech Lattice A General Setting for Construction A, and Quebbemann's 64-Dimensional Lattice Lattices Over Z[e'" 4 ], and Quebbemann's 32-Dimensional Lattice McKay's 40-Dimensional Extremal Lattice Repeated Differences and Craig's Lattices Lattices from Algebraic Number Theory Introduction Lattices from the Trace Norm Examples from Cyclotomic Fields Lattices from Class Field Towers Unimodular Lattices with an Automorphism of Prime Order Constructions D and D' Construction D Examples Construction D' Construction E Examples of Construction E 238 Chapter 9 Bounds for Codes and Sphere Packings N.J.A. Sloane Introduction Zonal Spherical Functions The 2-Point-Homogeneous Spaces Representations of G Zonal Spherical Functions Positive-Definite Degenerate Kernels The Linear Programming Bounds Codes and Their Distance Distributions The Linear Programming Bounds Bounds for Error-Correcting Codes Bounds for Constant-Weight Codes Bounds for Spherical Codes and Sphere Packings Other Bounds 265 Chapter 10 Three Lectures on Exceptional Groups J.H. Conway First Lecture Some Exceptional Behavior of the Groups L n (q) The Case p = The Case p = TheCasep = 7 269

7 lxix Contents 1.5 TheCase/?=ll A Presentation for M l Janko's Group of Order Second Lecture The Mathieu Group M The Stabilizer of an Octad The Structure of the Golay Code < The Structure of P (D.)K The Maximal Subgroups of M The Structure of P(Q) Third Lecture The Group Co 0 = 0 and Some of its Subgroups The Geometry of the Leech Lattice The Group 0 and its Subgroup N Subgroups of The Higman-Sims and McLaughlin Groups The Group Co, = Involutions in Congruences for Theta Series A Connection Between -0 and Fischer's Group Fi 2i 295 Appendix: On the Exceptional Simple Groups 296 Chapter 11 The Golay Codes and the Mathieu Groups J.H. Conway Introduction Definitions of the Hexacode Justification of a Hexacodeword Completing a Hexacodeword The Golay Code % lt and the MOG Completing Octads from 5 of their Points The Maximal Subgroups of M The Projective Subgroup L 2 (23) The Sextet Group 2 6 :3S The Octad Group 2 4 : A The Triad Group and the Projective Plane of Order The Trio Group 2": (S 3 x L 2 (7)) The Octern Group The Mathieu Group M 2, The Group M 22 : The Group M I2, the Tetracode and the MINIMOG Playing Cards and Other Games Further Constructions for M Chapter 12 A Characterization of the Leech Lattice J.H. Conway 331

8 Contents lxx Chapter 13 Bounds on Kissing Numbers A.M. Odlyzko and N.J.A. Sloane A General Upper Bound Numerical Results 338 Chapter 14 Uniqueness of Certain Spherical Codes E. Bannal and N.J.A. Sloane Introduction Uniqueness of the Code of Size 240 in fl Uniqueness of the Code of Size 56 in ft Uniqueness of the Code of Size in fl Uniqueness of the Code of Size 4600 in n Chapter 15 On the Classification of Integral Quadratic Forms J.H. Conway and N.J.A. Sloane Introduction Definitions Quadratic Forms Forms and Lattices; Integral Equivalence The Classification of Binary Quadratic Forms Cycles of Reduced Forms Definite Binary Forms Indefinite Binary Forms Composition of Binary Forms Genera and Spinor Genera for Binary Forms The p-adic Numbers The p-adic Numbers p-adic Square Classes An Extended Jacobi-Legendre Symbol Diagonalization of Quadratic Forms Rational Invariants of Quadratic Forms Invariants and the Oddity Formula Existence of Rational Forms with Prescribed Invariants The Conventional Form of the Hasse-Minkowski Invariant The Invariance and Completeness of the Rational Invariants The p-adic Invariants for Binary Forms The p-adic Invariants for «-Ary Forms The Proof of Theorem The Genus and its Invariants p-adic Invariants The p-adic Symbol for a Form 379

9 lxxi Contents Adic Invariants The 2-Adic Symbol Equivalences Between Jordan Decompositions A Canonical 2-Adic Symbol Existence of Forms with Prescribed Invariants A Symbol for the Genus Classification of Forms of Small Determinant and of p-elementary Forms Forms of Small Determinant p-elementary Forms The Spinor Genus Introduction The Spinor Genus Identifying the Spinor Kernel Naming the Spinor Operators for the Genus of/ Computing the Spinor Kernel from the p-ad\c Symbols Tractable and Irrelevant Primes When is There Only One Class in the Genus? The Classification of Positive Definite Forms Minkowski Reduction The Kneser Gluing Method Positive Definite Forms of Determinant 2 and Computational Complexity 402 Chapter 16 Enumeration of Unimodular Lattices J.H. Conway and N.J.A. Sloane The Niemeier Lattices and the Leech Lattice The Mass Formulae for Lattices Verifications of Niemeier's List The Enumeration of Unimodular Lattices in Dimensions n =s Chapter 17 The 24-Dimensional Odd Unimodular Lattices R.E. Borcherds 421 Chapter 18 Even Unimodular 24-Dimensional Lattices B.B. Venkov Introduction Possible Configurations of Minimal Vectors On Lattices with Root Systems of Maximal Rank Construction of the Niemeier Lattices A Characterization of the Leech Lattice 439

10 Contents lxxii Chapter 19 Enumeration of Extremal Self-Dual Lattices J.H. Conway, A.M. Odlyzko and N.J.A. Sloane Dimensions Dimensions Dimensions n s= Chapter 20 Finding the Closest Lattice Point J.H. Conway and N.J.A. Sloane Introduction The Lattices Z", D n and A Decoding Unions of Cosets "Soft Decision" Decoding for Binary Codes Decoding Lattices Obtained from Construction A Decoding Chapter 21 Voronoi Cells of Lattices and Quantization Errors J.H. Conway and N.J.A. Sloane Introduction Second Moments of Polytopes A Dirichlet's Integral B Generalized Octahedron or Crosspolytope C The n-sphere D ^-Dimensional Simplices E Regular Simplex F Volume and Second Moment of a Polytope in Terms of its Faces G Truncated Octahedron H Second Moment of Regular Polytopes Regular Polygons J Icosahedron and Dodecahedron K The Exceptional 4-Dimensional Polytopes Voronoi Cells and the Mean Squared Error of Lattice Quantizers A The Voronoi Cell of a Root Lattice B Voronoi Cell for A a C Voronoi Cell for D n (n s 4) D Voronoi Cells for, E Voronoi Cell for D* F Voronoi Cell for A* G The Walls of the Voronoi Cell 476 Chapter 22 A Bound for the Covering Radius of the Leech Lattice S.P. Norton 478

11 lxxiii Contents Chapter 23 The Covering Radius of the Leech Lattice J.H. Conway, R.A. Parker and N.J.A. Sloane Introduction The Coxeter-Dynkin Diagram of a Hole Holes Whose Diagram Contains an A n Subgraph Holes Whose Diagram Contains a D n Subgraph Holes Whose Diagram Contains an E n Subgraph 504 Chapter 24 Twenty-Three Constructions for the Leech Lattice J.H. Conway and N.J.A. Sloane The "Holy Constructions" The Environs of a Deep Hole 512 Chapter 25 The Cellular Structure of the Leech Lattice R.E. Borcherds, J.H. Conway and L. Queen Introduction Names for the Holes The Volume Formula The Enumeration of the Small Holes 521 Chapter 26 Lorentzian Forms for the Leech Lattice J.H. Conway and N.J.A. Sloane The Unimodular Lorentzian Lattices Lorentzian Constructions for the Leech Lattice 525 Chapter 27 The Automorphism Group of the 26-Dimensional Even Unimodular Lorentzian Lattice J.H. Conway Introduction The Main Theorem 530 Chapter 28 Leech Roots and Vinberg Groups J.H. Conway and N.J.A. Sloane The Leech Roots Enumeration of the Leech Roots The Lattices I nl for «s Vinberg's Algorithm and the Initial Batches of Fundamental Roots The Later Batches of Fundamental Roots 552

12 Contents lxxiv Chapter 29 The Monster Group and its Dimensional Space J.H. Conway Introduction The Golay Code < and the Parker Loop 9> The Mathieu Group A/ 24 ; the Standard Automorphisms of The Golay Cocode %* and the Diagonal Automorphisms The Group N of Triple Maps The kernel K and the Homomorphism g^> g The Structures of Various Subgroups of N The Leech Lattice A 24 and the Group Q x Short Elements The Basic Representations of N The Dictionary The Algebra._^ The Definition of the Monster Group G, and its Finiteness Identifying the Monster 564 Appendix 1. Computing in 3> 565 Appendix 2. A Construction for Appendix 3. Some Relations in Q, 566 Appendix 4. Constructing Representations for N, 568 Appendix 5. Building the Group G, 569 Chapter 30 A Monster Lie Algebra? R.E. Borcherds, J.H. Conway, L. Queen and N.J.A. Sloane 570 Bibliography 574 Supplementary Bibliography 642 Index 681