Codes on Euclidean spheres [electronic resource] / Thomas Ericson, Victor Zinoviev.

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.

Includes bibliographical references (p. 519-540) and index.

Description based on print version record.

Cover -- Contents -- Chapter 1. Introduction -- 1.1 Definitions and basic properties -- 1.2 Examples of spherical codes -- 1.3 Two basic functions -- 1.4 The Rankin bounds -- 1.5 The Simplex and the Biorthogonal codes -- 1.6 The Chabauty-Shannon-Wyner bound -- 1.7 The direct sum -- Chapter 2. The linear programming bound -- 2.1 Introduction -- 2.2 Spherical polynomials -- 2.3 The linear programming bound -- 2.4 Orthogonal polynomials -- 2.5 The Levenshtein bound -- 2.6 The Boyvalenkov-Danev-Bumova criterion -- 2.7 Properties of the Levenshtein bound -- Chapter 3. Codes in dimension n=3 -- 3.1 Introduction -- 3.2 The optimal codes -- 3.3 Additional comments -- 3.4 The Fejes T�oth bound -- 3.5 Optimality in the case M=7 -- 3.6 The Coxeter-B�or�oczky extension -- 3.7 Thirteen spheres -- Chapter 4. Permutation codes -- 4.1 Introduction -- 4.2 Variant 1 -- 4.3 Best variant 1 codes -- 4.4 Variant 2a -- 4.5 Variant 2b -- 4.6 Dimensionality -- 4.7 Decoding -- 4.8 General comments -- Chapter 5. Symmetric alphabets -- 5.1 Introduction -- 5.2 An introductory example -- 5.3 Binary labeling -- 5.4 The construction. 2 K 4 -- 5.5 The construction: general case -- 5.6 A simple example -- 5.7 Analysis -- 5.8 Explicit constructions -- 5.9 Unions -- 5.10 Extensions -- 5.11 Concluding remarks -- Chapter 6. Non-symmetric alphabets -- 6.1 Introduction -- 6.2 The binary balanced mapping -- 6.3 Comments -- 6.4 Unions from the CW2-construction -- 6.5 Non-symmetric ternary alphabet -- 6.6 The general balanced construction -- Chapter 7. Polyphase codes -- 7.1 Introduction -- 7.2 General properties -- 7.3 The case q = 3 -- 7.4 The case q = 4 -- 7.5 The case q = 6 -- 7.6 The case q = 8 -- 7.7 Two special constructions -- 7. 8 A general comment -- Chapter 8. Group codes -- 8.1 Introduction -- 8.2 Basic properties -- 8.3 Groups represented by matrices -- 8.4 Group codes in binary Hamming spaces -- 8.5 Group codes from binary codes -- 8.6 Dual codes and MacWilliams' identity -- 8.7 Finite reflection groups -- 8.8 Codes from finite reflection groups -- 8.9 Examples -- 8.10 Remarks on some specific codes -- Chapter 9. Distance regular spherical codes -- 9.1 Introduction -- 9.2 Association schemes -- 9.3 Metric schemes -- 9.4 Strongly regular graphs -- 9.5 The absolute bound -- 9.6 Spherical designs -- 9.7 Regular polytopes -- Chapter 10. Lattices -- 10.1 Introduction -- 10.2 Lattices -- 10.3 The root lattices -- 10.4 Sphere packings and packing bounds -- 10.5 Sphere packings and codes -- 10.6 Lattices and codes -- 10.7 Expurgated constructions -- 10.8 The Leech lattice -- 10.9 Theta functions -- 10.10 Spherical codes from lattices -- 10.11 Theta function.