J.H. Conway N.J.A. Sloane

Sphere Packings,Lattices and Groups

Third Edition

With Additional Contributions byE. 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

Contents

Preface to First Edition vPreface to Third Edition xvList of Symbols lxi

Chapter 1Sphere Packings and Kissing NumbersJ.H. Conway and N.J.A. Sloane 1

The Sphere Packing Problem II Packing Ball Bearings 1

.2 Lattice Packings 33 Nonlattice Packings 74 /7-Dimensional Packings 85 Sphere Packing Problem—Summary of Results 12

2. The Kissing Number Problem 212.1 The Problem of the Thirteen Spheres 212.2 Kissing Numbers in Other Dimensions 212.3 Spherical Codes 242.4 The Construction of Spherical Codes from Sphere

Packings 262.5 The Construction of Spherical Codes from Binary Codes . . . 262.6 Bounds on A(/i,«j>) 27

Appendix: Planetary Perturbations 29

Chapter 2Coverings, Lattices and QuantizersJ.H. Conway and N.J.A. Sloane 31

1. The Covering Problem 311.1 Covering Space with Overlapping Spheres 311.2 The Covering Radius and the Voronoi Cells 331.3 Covering Problem—Summary of Results 361.4 Computational Difficulties in Packings and Coverings 40

lxv Contents

2. Lattices, Quadratic Forms and Number Theory 412.1 The Norm of a Vector 412.2 Quadratic Forms Associated with a Lattice 422.3 Theta Series and Connections with Number Theory 442.4 Integral Lattices and Quadratic Forms 472.5 Modular Forms 502.6 Complex and Quaternionic Lattices 523. Quantizers 563.1 Quantization, Analog-to-Digital Conversion and Data

Compression 563.2 The Quantizer Problem 593.3 Quantizer Problem—Summary of Results 59

Chapter 3Codes, Designs and GroupsJ.H. Conway and N.J.A. Sloane 63

1. The Channel Coding Problem 631.1 The Sampling Theorem 631.2 Shannon's Theorem 661.3 Error Probability 691.4 Lattice Codes for the Gaussian Channel 712. Error-Correcting Codes 752.1 The Error-Correcting Code Problem 752.2 Further Definitions from Coding Theory 772.3 Repetition, Even Weight and Other Simple Codes 792.4 Cyclic Codes 792.5 BCH and Reed-Solomon Codes 812.6 Justesen Codes 822.7 Reed-Muller Codes 832.8 Quadratic Residue Codes 842.9 Perfect Codes 852.10 The Pless Double Circulant Codes 862.11 Goppa Codes and Codes from Algebraic Curves 872.12 Nonlinear Codes 872.13 Hadamard Matrices 873. r-Designs, Steiner Systems and Spherical f-Designs 883.1 /-Designs and Steiner Systems 883.2 Spherical r-Designs 894. The Connections with Group Theory 904.1 The Automorphism Group of a Lattice 904.2 Constructing Lattices and Codes from Groups 92

Chapter 4Certain Important Lattices and Their PropertiesJ.H. Conway and N.J.A. Sloane 94

1. Introduction 942. Reflection Groups and Root Lattices 953. Gluing Theory 99

Contents lxvi

4. Notation; Theta Functions 1014.1 Jacobi Theta Functions 1025. The ^-Dimensional Cubic Lattice Z" 1066. The ^-Dimensional Lattices Aa and A* 1086.1 The Lattice An 1086.2 The Hexagonal Lattice 1106.3 The Face-Centered Cubic Lattice 1126.4 The Tetrahedral or Diamond Packing 1136.5 The Hexagonal Close-Packing 1136.6 The. Dual Lattice A* 1156.7 The Body-Centered Cubic Lattice 1167. The H-Dimensional Lattices £>„ and D* 1177.1 The Lattice Dn 1177.2 The Four-Dimensional Lattice D4 1187.3 The Packing Dn 1197.4 The Dual Lattice D* 1208. The Lattices Eb, E7 and £8 1208.1 The 8-Dimensional Lattice £„ 1208.2 The 7-Dimensional Lattices £7 and E? 1248.3 The 6-Dimensional Lattices £6 and Et 1259. The 12-Dimensional Coxeter-Todd Lattice Kn 127

10. The 16-Dimensional Barnes-Wall Lattice A,6 12911. The 24-Dimensional Leech Lattice A24 131

Chapter 5Sphere Packing and Error-Correcting CodesJ. Leech and N.J.A. Sloane 136

1. Introduction 1361.1 The Coordinate Array of a Point 1372. Construction A 1372.1 The Construction 1372.2 Center Density 1372.3 Kissing Numbers 1382.4 Dimensions 3 to 6 1382.5 Dimensions 7 and 8 1382.6 Dimensions 9 to 12 1392.7 Comparison of Lattice and Nonlattice Packings 1403. Construction B 1413.1 The Construction 1413.2 Center Density and Kissing Numbers 1413.3 Dimensions 8, 9 and 12 1423.4 Dimensions 15 to 24 1424. Packings Built Up by Layers 1424.1 Packing by Layers 1424.2 Dimensions 4 to 7 1444.3 Dimensions 11 and 13 to 15 1444.4 Density Doubling and the Leech Lattice A24 1454.5 Cross Sections of A24 1455. Other Constructions from Codes 146

lxvii Contents

5.1 A Code of Length 40 1465.2 A Lattice Packing in R40 1475.3 Cross Sections of A40 1485.4 Packings Based on Ternary Codes 1485.5 Packings Obtained from the Pless Codes 1485.6 Packings Obtained from Quadratic Residue Codes 1495.7 Density Doubling in R24 and R48 1496. Construction C 1506.1 The Construction 1506.2 Distance Between Centers 1506.3 Center Density 1506.4 Kissing Numbers 1516.5 Packings Obtained from Reed-Muller Codes 1516.6 Packings Obtained from BCH and Other Codes 1526.7 Density of BCH Packings 1536.8 Packings Obtained from Justesen Codes 155

Chapter 6Laminated LatticesJ.H. Conway and N.J.A. Sloane 157

1. Introduction 1572. The Main Results 1633. Properties of Ao to A8 1684. Dimensions 9 to 16 1705. The Deep Holes in A,6 1746. Dimensions 17 to 24 1767. Dimensions 25 to 48 177

Appendix: The Best Integral Lattices Known 179

Chapter 7Further Connections Between Codes and Latt icesN.J.A. Sloane 181

1. Introduction 1812. Construction A 1823. Self-Dual (or Type I) Codes and Lattices .: 1854. Extremal Type I Codes and Lattices 1895. ConstructionB 1916. Type II Codes and Lattices 1917. Extremal Type II Codes and Lattices 1938. Constructions A and B for Complex Lattices 1979. Self-Dual Nonbinary Codes and Complex Lattices 202

10. Extremal Nonbinary Codes and Complex Lattices 205

Chapter 8Algebraic Construct ions for Latt icesJ.H. Conway and N.J.A. Sloane 206

1. Introduction 2062. The Icosians and the Leech Lattice 207

Contents lxviii

2.1 The Icosian Group 2072.2 The Icosian and Turyn-Type Constructions for the Leech

Lattice 2103. A General Setting for Construction A, and Quebbemann's

64-Dimensional Lattice 2114. Lattices Over Z[e'"4], and Quebbemann's 32-Dimensional

Lattice 2155. McKay's 40-Dimensional Extremal Lattice 2216. Repeated Differences and Craig's Lattices 2227. Lattices from Algebraic Number Theory 2247.1 Introduction 2247.2 Lattices from the Trace Norm 2247.3 Examples from Cyclotomic Fields 2277.4 Lattices from Class Field Towers 2277.5 Unimodular Lattices with an Automorphism of Prime

Order 2298. Constructions D and D' 2328.1 Construction D 2328.2 Examples 2338.3 Construction D' 2359. Construction E 236

10. Examples of Construction E 238

Chapter 9Bounds for Codes and Sphere PackingsN.J.A. Sloane 245

1. Introduction 2452. Zonal Spherical Functions 2492.1 The 2-Point-Homogeneous Spaces 2502.2 Representations of G 2522.3 Zonal Spherical Functions 2532.4 Positive-Definite Degenerate Kernels 2563. The Linear Programming Bounds 2573.1 Codes and Their Distance Distributions 2573.2 The Linear Programming Bounds 2583.3 Bounds for Error-Correcting Codes 2603.4 Bounds for Constant-Weight Codes 2633.5 Bounds for Spherical Codes and Sphere Packings 2634. Other Bounds 265

Chapter 10Three Lectures on Exceptional GroupsJ.H. Conway 267

1. First Lecture 2671.1 Some Exceptional Behavior of the Groups Ln(q) 2671.2 The Case p = 3 2691.3 The Case p = 5 2691.4 TheCasep = 7 269

lxix Contents

1.5 T h e C a s e / ? = l l 2711.6 A Presentation for Ml2 2731.7 Janko's Group of Order 175560 2732. Second Lecture 2742.1 The Mathieu Group M24 2742.2 The Stabilizer of an Octad 2762.3 The Structure of the Golay Code <€24 2782.4 The Structure of P (D.)K24 2782.5 The Maximal Subgroups of M24 2792.6 The Structure of P(Q) 2833. Third Lecture 2863.1 The Group Co0 = 0 and Some of its Subgroups 2863.2 The Geometry of the Leech Lattice 2863.3 The Group 0 and its Subgroup N 2873.4 Subgroups of 0 2903.5 The Higman-Sims and McLaughlin Groups 2923.6 The Group Co, = -3 2933.7 Involutions in -0 2943.8 Congruences for Theta Series 2943.9 A Connection Between -0 and Fischer's Group Fi2i 295

Appendix: On the Exceptional Simple Groups 296

Chapter 11The Golay Codes and the Mathieu GroupsJ.H. Conway 299

1. Introduction 2992. Definitions of the Hexacode 3003. Justification of a Hexacodeword 3024. Completing a Hexacodeword 3025. The Golay Code %lt and the MOG 3036. Completing Octads from 5 of their Points 3057. The Maximal Subgroups of M24 3078. The Projective Subgroup L2(23) 3089. The Sextet Group 26:3S6 309

10. The Octad Group 24: A8 31111. The Triad Group and the Projective Plane of Order 4 31412. The Trio Group 2": (S3 x L2(7)) 31613. The Octern Group 31814. The Mathieu Group M2, 31915. The Group M22:2 31916. The Group MI2, the Tetracode and the MINIMOG 32017. Playing Cards and Other Games 32318. Further Constructions for M12 327

Chapter 12A Characterization of the Leech LatticeJ.H. Conway 331

Contents lxx

Chapter 13Bounds on Kissing NumbersA.M. Odlyzko and N.J.A. Sloane 337

1. A General Upper Bound 3372. Numerical Results 338

Chapter 14Uniqueness of Certain Spherical CodesE. Bannal and N.J.A. Sloane 340

1. Introduction 3402. Uniqueness of the Code of Size 240 in fl8 3423. Uniqueness of the Code of Size 56 in ft7 3444. Uniqueness of the Code of Size 196560 in fl24 3455. Uniqueness of the Code of Size 4600 in n23 349

Chapter 15On the Classification of Integral Quadratic FormsJ.H. Conway and N.J.A. Sloane 352

1. Introduction 3522. Definitions 3542.1 Quadratic Forms 3542.2 Forms and Lattices; Integral Equivalence 3553. The Classification of Binary Quadratic Forms 3563.1 Cycles of Reduced Forms 3563.2 Definite Binary Forms 3573.3 Indefinite Binary Forms 3593.4 Composition of Binary Forms 3643.5 Genera and Spinor Genera for Binary Forms 3664. The p-Adic Numbers 3664.1 The p-Adic Numbers 3674.2 p-Adic Square Classes 3674.3 An Extended Jacobi-Legendre Symbol 3684.4 Diagonalization of Quadratic Forms 3695. Rational Invariants of Quadratic Forms 3705.1 Invariants and the Oddity Formula 3705.2 Existence of Rational Forms with Prescribed Invariants 3725.3 The Conventional Form of the Hasse-Minkowski

Invariant 3736. The Invariance and Completeness of the Rational

Invariants 3736.1 The p-Adic Invariants for Binary Forms 3736.2 The p-Adic Invariants for «-Ary Forms 3756.3 The Proof of Theorem 7 3777. The Genus and its Invariants 3787.1 p-Adic Invariants 3787.2 The p-Adic Symbol for a Form 379

lxxi Contents

7.3 2-Adic Invariants 3807.4 The 2-Adic Symbol 3807.5 Equivalences Between Jordan Decompositions 3817.6 A Canonical 2-Adic Symbol 3827.7 Existence of Forms with Prescribed Invariants 3827.8 A Symbol for the Genus 3848. Classification of Forms of Small Determinant and of

p-Elementary Forms 3858.1 Forms of Small Determinant 3858.2 p-Elementary Forms 3869. The Spinor Genus 3889.1 Introduction 3889.2 The Spinor Genus 3899.3 Identifying the Spinor Kernel 3909.4 Naming the Spinor Operators for the Genus of/ 3909.5 Computing the Spinor Kernel from the p-Ad\c Symbols 3919.6 Tractable and Irrelevant Primes 3929.7 When is There Only One Class in the Genus? 393

10. The Classification of Positive Definite Forms 39610.1 Minkowski Reduction 39610.2 The Kneser Gluing Method 39910.3 Positive Definite Forms of Determinant 2 and 3 39911. Computational Complexity 402

Chapter 16Enumeration of Unimodular LatticesJ.H. Conway and N.J.A. Sloane 406

1. The Niemeier Lattices and the Leech Lattice 4062. The Mass Formulae for Lattices 4083. Verifications of Niemeier's List 4104. The Enumeration of Unimodular Lattices in Dimensions

n =s 23 413

Chapter 17The 24-Dimensional Odd Unimodular LatticesR.E. Borcherds 421

Chapter 18Even Unimodular 24-Dimensional LatticesB.B. Venkov 429

1. Introduction 4292. Possible Configurations of Minimal Vectors 4303. On Lattices with Root Systems of Maximal Rank 4334. Construction of the Niemeier Lattices 4365. A Characterization of the Leech Lattice 439

Contents lxxii

Chapter 19Enumeration of Extremal Self-Dual LatticesJ.H. Conway, A.M. Odlyzko and N.J.A. Sloane 441

1. Dimensions 1-16 4412. Dimensions 17-47 4413. Dimensions n s= 48 443

Chapter 20Finding the Closest Lattice PointJ.H. Conway and N.J.A. Sloane 445

1. Introduction 4452. The Lattices Z", Dn and A 4463. Decoding Unions of Cosets 4484. "Soft Decision" Decoding for Binary Codes 4495. Decoding Lattices Obtained from Construction A 4506. Decoding £8 450

Chapter 21Voronoi Cells of Lattices and Quantization ErrorsJ.H. Conway and N.J.A. Sloane 451

1. Introduction 4512. Second Moments of Polytopes 4532.A Dirichlet's Integral 4532.B Generalized Octahedron or Crosspolytope 4542.C The n-Sphere 4542.D ^-Dimensional Simplices 4542.E Regular Simplex 4552.F Volume and Second Moment of a Polytope in Terms

of its Faces 4552.G Truncated Octahedron 4562.H Second Moment of Regular Polytopes 4562.1 Regular Polygons 4572.J Icosahedron and Dodecahedron 4572.K The Exceptional 4-Dimensional Polytopes 4573. Voronoi Cells and the Mean Squared Error of Lattice

Quantizers 4583. A The Voronoi Cell of a Root Lattice 4583.B Voronoi Cell for Aa 4613.C Voronoi Cell for Dn (n s 4) 4643.D Voronoi Cells for £„,£„£„ 4643.E Voronoi Cell for D* 4653.F Voronoi Cell for A* 4743.G The Walls of the Voronoi Cell 476

Chapter 22A Bound for the Covering Radius of the Leech LatticeS.P. Norton 478

lxxiii Contents

Chapter 23The Covering Radius of the Leech LatticeJ.H. Conway, R.A. Parker and N.J.A. Sloane 480

1. Introduction 4802. The Coxeter-Dynkin Diagram of a Hole 4823. Holes Whose Diagram Contains an An Subgraph 4864. Holes Whose Diagram Contains a Dn Subgraph 4975. Holes Whose Diagram Contains an En Subgraph 504

Chapter 24Twenty-Three Constructions for the Leech LatticeJ.H. Conway and N.J.A. Sloane 508

1. The "Holy Constructions" 5082. The Environs of a Deep Hole 512

Chapter 25The Cellular Structure of the Leech LatticeR.E. Borcherds, J.H. Conway and L. Queen 515

1. Introduction 5152. Names for the Holes 5153. The Volume Formula 5164. The Enumeration of the Small Holes 521

Chapter 26Lorentzian Forms for the Leech LatticeJ.H. Conway and N.J.A. Sloane 524

1. The Unimodular Lorentzian Lattices 5242. Lorentzian Constructions for the Leech Lattice 525

Chapter 27The Automorphism Group of the 26-Dimensional EvenUnimodular Lorentzian LatticeJ.H. Conway 529

1. Introduction 5292. The Main Theorem 530

Chapter 28Leech Roots and Vinberg GroupsJ.H. Conway and N.J.A. Sloane 534

1. The Leech Roots 5342. Enumeration of the Leech Roots 5433. The Lattices Inlfor « s 19 5494. Vinberg's Algorithm and the Initial Batches

of Fundamental Roots 5495. The Later Batches of Fundamental Roots 552

Contents lxxiv

Chapter 29The Monster Group and its 196884-Dimensional SpaceJ.H. Conway 556

1. Introduction 5562. The Golay Code <€ and the Parker Loop 9> 5583. The Mathieu Group A/24; the Standard Automorphisms

of 9 5584. The Golay Cocode %* and the Diagonal Automorphisms ... 5585. The Group N of Triple Maps 5596. The kernel K and the Homomorphism g^> g 5597. The Structures of Various Subgroups of N 5598. The Leech Lattice A24 and the Group Qx 5609. Short Elements 561

10. The Basic Representations of N 56111. The Dictionary 56212. The Algebra ._̂ 56313. The Definition of the Monster Group G, and its

Finiteness 56314. Identifying the Monster 564

Appendix 1. Computing in 3> 565Appendix 2. A Construction for 9 565Appendix 3. Some Relations in Q, 566Appendix 4. Constructing Representations for N, 568Appendix 5. Building the Group G, 569

Chapter 30A Monster Lie Algebra?R.E. Borcherds, J.H. Conway, L. Queen andN.J.A. Sloane 570

Bibliography 574

Supplementary Bibliography 642

Index 681