Codes on Euclidean SpheresCodes 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 errorcontrol in communication systems. In that context the use of spherical codes is often referred to as "coded modulation."

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Contents
1  
27  
Chapter
3 Codes in dimension n3  67 
Chapter
4 Permutation codes  107 
Chapter
5 Symmetric alphabets  129 
Chapter
6 Nonsymmetric alphabets  179 
Chapter
7 Polyphase codes  195 
Chapter
8 Group codes  205 
Chapter
10 Lattices  337 
Chapter
11 Decoding  389 
Appendix A Algebraic codes and designs  417 
Appendix B Spheres in R n  439 
Appendix C Spherical geometry  443 
Appendix D Tables  451 
Bibliography  519 
541  
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Common terms and phrases
algorithm alphabet antipodal association scheme binary codes Biorthogonal code Chapter code A2 code with parameters codes of dimension codewords components connected Consider constant weight code construction coordinates corresponding Coxeter decoding define distance regular edges eigenvalues elements equality Example finite reflection group formula function Gegenbauer polynomials given Gram matrix group code Hamming Hamming code icosahedron implies indicated inequality initial vector inner product integer known lattice Lemma Levenshtein bound linear code mapping notation notice obtained by Y3 optimal code orthogonal pair parameters n,p partition points possible Proof reflection group regular polytope respectively result root satisfies sequence simple Simplex code space sphere packing spherical code squared distance squared minimum distance Steiner system strongly regular graph subcode subset tail code Theorem transformations triangle trivial extension trivial left union of 31 US(PA US(PA(A US(PB valency Variant vertex figures vertices Y3 from C1 zeros
Popular passages
Page 535  A New Multilevel Coding Method Using Error Correcting Codes," IEEE Trans, on Information Theory, Vol. IT23, No. 3, pp. 371376, May 1977. 5. VV Ginzburg, "Multidimensional Signals for a Continuous Channel," Problemy Peredachi Informatsii, Vol. 20, No. 1, pp. 2846, 1984. 6. SI Sayegh, "A Class of Optimum Block Codes in Signal Space," IEEE Trans, on Communications, Vol.
Page 535  Analytical treatment of the polytopes regularly derived from the regular polytopes", Ver.