Elsevier S & T

Codes on Euclidean Spheres

Ericson, T.; Zinoviev, V.

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Table of contents
  • Contentsix
  • Chapter 1. Introduction1
  • 1.1 Definitions and basic properties1
  • 1.2 Examples of spherical codes5
  • 1.3 Two basic functions12
  • 1.4 The Rankin bounds14
  • 1.5 The Simplex and the Biorthogonal codes17
  • 1.6 The Chabauty–Shannon–Wyner bound19
  • 1.7 The direct sum24
  • Chapter 2. The linear programming bound27
  • 2.1 Introduction27
  • 2.2 Spherical polynomials28
  • 2.3 The linear programming bound39
  • 2.4 Orthogonal polynomials42
  • 2.5 The Levenshtein bound47
  • 2.6 The Boyvalenkov–Danev–Bumova criterion58
  • 2.7 Properties of the Levenshtein bound62
  • Chapter 3. Codes in dimension n=367
  • 3.1 Introduction67
  • 3.2 The optimal codes68
  • 3.3 Additional comments79
  • 3.4 The Fejes Tóth bound81
  • 3.5 Optimality in the case M=786
  • 3.6 The Coxeter–Böröczky extension97
  • 3.7 Thirteen spheres98
  • Chapter 4. Permutation codes107
  • 4.1 Introduction107
  • 4.2 Variant 1108
  • 4.3 Best variant 1 codes113
  • 4.4 Variant 2a116
  • 4.5 Variant 2b119
  • 4.6 Dimensionality121
  • 4.7 Decoding123
  • 4.8 General comments126
  • Chapter 5. Symmetric alphabets129
  • 5.1 Introduction129
  • 5.2 An introductory example130
  • 5.3 Binary labeling134
  • 5.4 The construction. 2 ≥ K ≥ 4138
  • 5.5 The construction: general case142
  • 5.6 A simple example149
  • 5.7 Analysis151
  • 5.8 Explicit constructions156
  • 5.9 Unions161
  • 5.10 Extensions169
  • 5.11 Concluding remarks175
  • Chapter 6. Non-symmetric alphabets179
  • 6.1 Introduction179
  • 6.2 The binary balanced mapping179
  • 6.3 Comments182
  • 6.4 Unions from the CW2-construction183
  • 6.5 Non-symmetric ternary alphabet185
  • 6.6 The general balanced construction188
  • Chapter 7. Polyphase codes195
  • 7.1 Introduction195
  • 7.2 General properties195
  • 7.3 The case q = 3197
  • 7.4 The case q = 4199
  • 7.5 The case q = 6200
  • 7.6 The case q = 8200
  • 7.7 Two special constructions201
  • 7. 8 A general comment202
  • Chapter 8. Group codes205
  • 8.1 Introduction205
  • 8.2 Basic properties206
  • 8.3 Groups represented by matrices210
  • 8.4 Group codes in binary Hamming spaces213
  • 8.5 Group codes from binary codes218
  • 8.6 Dual codes and MacWilliams' identity223
  • 8.7 Finite reflection groups230
  • 8.8 Codes from finite reflection groups242
  • 8.9 Examples247
  • 8.10 Remarks on some specific codes252
  • Chapter 9. Distance regular spherical codes257
  • 9.1 Introduction257
  • 9.2 Association schemes263
  • 9.3 Metric schemes275
  • 9.4 Strongly regular graphs285
  • 9.5 The absolute bound311
  • 9.6 Spherical designs314
  • 9.7 Regular polytopes324
  • Chapter 10. Lattices337
  • 10.1 Introduction337
  • 10.2 Lattices337
  • 10.3 The root lattices344
  • 10.4 Sphere packings and packing bounds348
  • 10.5 Sphere packings and codes353
  • 10.6 Lattices and codes358
  • 10.7 Expurgated constructions362
  • 10.8 The Leech lattice365
  • 10.9 Theta functions367
  • 10.10 Spherical codes from lattices371
  • 10.11 Theta functions for expurgated constructions378
  • 10.12 Unions of shells382
  • Chapter 11. Decoding389
  • 11.1 Introduction389
  • 11.2 The problem390
  • 11.3 Preliminaries391
  • 11.4 Generalized minimum distance decoding395
  • 11.5 The Chase decoder400
  • 11.6 Parallel decoding404
  • 11.7 Decoding Y3409
  • Appendix A. Algebraic codes and designs417
  • Appendix B. Spheres in R n439
  • Appendix C. Spherical geometry443
  • Appendix D. Tables451
  • D.1 Introduction451
  • D.2 Notation452
  • D.3 Spherical codes of dimension n = 3455
  • D.4 Spherical codes of dimension n = 4456
  • D.5 Spherical codes of dimension n = 5456
  • D.6 Spherical codes of dimension n = 6457
  • D.7 Spherical codes of dimension n = 7460
  • D.8 Spherical codes of dimension n = 8463
  • D.9 Spherical codes of dimension n = 9465
  • D.10 Spherical codes of dimension n = 10468
  • D.11 Spherical codes of dimension n = 11472
  • D.12 Spherical codes of dimension n = 12475
  • D.13 Spherical codes of dimension n = 13477
  • D.14 Spherical codes of dimension n = 14480
  • D.15 Spherical codes of dimension n = 15483
  • D.16 Spherical codes of dimension n = 16485
  • D.17 Spherical codes of dimension n = 17488
  • D.18 Spherical codes of dimension n = 18492
  • D.19 Spherical codes of dimension n = 19497
  • D.20 Spherical codes of dimension n = 20501
  • D.21 Spherical codes of dimension n = 21506
  • D.22 Spherical codes of dimension n = 22508
  • D.23 Spherical codes of dimension n = 23512
  • D.24 Spherical codes of dimension n = 24517
  • Bibliography519
  • Index541
Book details
  • Vendor Elsevier S & T
  • SKU 9780444503299
  • ISBN-13 9780080502168
  • Author Ericson, T.; Zinoviev, V.
  • Category Mathematics
  • Subject Number Theory

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Description eBook delivery & access Reviews
Codes on Euclidean spheres are often referred to as spherical codes. They are of interest from mathematical, physical and engineering points of view. Mathematically the topic belongs to the realm of algebraic combinatorics, with close connections to number theory, geometry, combinatorial theory, and - of course - to algebraic coding theory. The connections to physics occur within areas like crystallography and nuclear physics. In engineering spherical codes are of central importance in connection with error-control in communication systems. In that context the use of spherical codes is often referred to as "coded modulation."


The book offers a first complete treatment of the mathematical theory of codes on Euclidean spheres. Many new results are published here for the first time. Engineering applications are emphasized throughout the text. The theory is illustrated by many examples. The book also contains an extensive table of best known spherical codes in dimensions 3-24, including exact constructions.