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182 Bibliography

Index

Es-Iattice, 14 t - (v, k,)..) design, 65

action group, 15

algebraic number field, 136 algebraically dependent, 60 algebraically independent, 60 Alpbach

theorem of, 149 automorphism, 103 automorphism group, 10 automorphism group of a symmetric

bilinear form module, 3 automorphism of symmetric bilinear

form modules, 3

basis of a lattice, 1 BCH bound, 79 BCH code, 78 Bernoulli numbers, 52 binary code, 6, 68 binary cube group, 160 binary Golay code, 69 binary tetrahedral group, 152 block, 65 bound

BCH, 79 BroU(~ and Enguehard

theorem of, 157

character, 96 code, 6

183

BCH, 78 binary, 6, 68 binary Golay, 69 cyclic, 78 doubly even, 9 dual,9 even weight, 67 extended, 11 extended binary Golay, 64, 69 extended Hamming, 11 extended ternary Golay, 145 extremal, 76 Hamming, 7 length of a, 6 linear, 8 minimum distance of a, 7 perfect, 66 puncturing of a, 67 quadratic residue, 82 Reed-Solomon, 79 RS,79 self-dual, 9 simplex, 80 ternary, 6 tetrad, 105 trivial, 6

codes equivalent, 10, 105

codewords, 6 commensurable, 168 Conway and Sloane

theorem of, 118, 130 Conway's theorem, 109, 132

184

Conway, Parker, and Sloane theorems of, 115

covering problem, 112 covering radius, 112 Coxeter number, 28, 116 Coxeter-Dynkin diagram, 14, 19 cusp form, 53 cyclic code, 78

decomposable, 17 deep hole, 113 Delaunay polytope, 112 dependent

algebraically, 60 design, 65

points of a, 65 diagram

Coxeter-Dynkin, 14, 19 extended Coxeter-Dynkin, 34 hole, 114 ordinarY., ,34

different, 167 direct sum

orthogonal, 5 discriminant, 4, 138, 169 discriminant quadratic form, 103 distance, 7 domain

fundamental, 40 doubly even code, 9 doubly transitive, 84 dual code, 9 dual ideal, 167 dual lattice, 2, 168 dual module, 4

Eisenstein series, 48 normalized, 52

Enguehard and Brow~ theorem of, 157

equation icosahedral, 164

tetrahedral, 158 equation of the cube, 159 equivalent, 40 equivalent codes, 10, 105 equivalent holes, 112 even, 5, 164 even lattice, 5, 140, 171 even overlattice, 103 even weight code, 67 exact sequence

split, 15

Index

extended binary Golay code, 64, 69 extended Coxeter-Dynkin diagram,

34 extended Hamming code, 11 extended ternary Golay code, 145 extremal code, 76 extremal lattice, 77 extremal theta function, 77 extremal weight enumerator, 76

field algebraic number, 136

finite quadratic form, 102 form

cusp, 53 Hilbert modular, 148 modular, 43

formula Poisson summation, 44

Fourier transform, 44 fractional ideal, 167 fractional ideals

product of, 167 function

extremal theta, 77 Riemann (-, 50 theta, 39, 169

fundamental domain, 40 fundamental parallelotope, 1 fundamental system of roots, 17

Index

Gaussian sums reciprocity law for, 96

generator matrix, 9 generator polynomial, 78 Gleason's theorem, 75 Gleason, Pierce, and Sloane

theorem of, 165 glue vector, 116 gluing, 116 Golay code, 64, 69

extended ternary, 145 group

automorphism, 10 binary tetrahedral, 152 Hilbert modular, 161 modular, 40 Weyl, 25

group action, 15

half-integral weight, 160 Hamming code, 7 Hamming distance, 7 Hamming weight enumerator, 62 harmonic polynomial, 30 hexacode, 68 highest root, 32 Hilbert modular form, 148

symmetric, 148 Hilbert modular group, 161 Hilbert modular surface, 161 Hirzebruch

theorem of, 164 Hirzebruch and van der Geer

theorem of, 149 hole, 112

deep, 113 shallow, 113

hole diagram, 114 holes

equivalent, 112

icosahedral equation, 164

ideal dual, 167 fractional, 167

ideals product of fractional, 167

idempotent, 81 identity

MacWilliams, 74

185

identity for Lee weight enumerators MacWilliams, 165

identity for ternary codes MacWilliams, 156

incidence matrix, 65 indecomposable, 17 independent

algebraically, 60 information rate, 7 integers, 136 integral, 3, 136 integral lattice, 3, 171 irreducible lattice, 22 isomorphic lattices, 103 isomorphic symmetric bilinear form

modules, 3 isotropic vector, 119

kissing number, 111 kissing number problem, 111 Klein's theorem, 164

lattice, 1, 167 basis of a, 1 dual, 2, 168 even, 5, 140, 171 extremal, 77 integral, 3, 171 irreducible, 22 Leech, 71 minimum squared distance of a,

111 Niemeier, 116 reducible, 22

186

root, 16 unimodular, 4

lattices isomorphic, 103

law for Gaussian sums reciprocity, 96

Lee weight enumerator, 149 Lee weight enumerators

MacWilliams identity for, 165 Leech lattice, 71 lenght function, 104 length, 6 level, 95, 172 linear code, 8

Mac Williams identity, 74 Mac Williams identity for Lee weight

enumerators, 165 Mac Williams identity for ternary

codes, 156 matrix

generator, 9 incidence, 65 parity check, 9

minimum distance, 7 minimum squared distance, 111 modular form, 43

Hilbert, 148 symmetric, 148

modular group, 40 Hilbert, 161

modular surface Hilbert, 161

module dual, 4 symmetric bilinear form, 3

monic polynomial, 77

negative root, 31 neighbours, 109 Newton number problem, 111 Niemeier lattice, 116

Index

Niemeier's theorem, 106 normalized Eisenstein series, 52 number

Coxeter, 28, 116 kissing, 111 Newton, 111

orbit, 40 order, 54 ordinary diagram, 34 orthogonal, 5 orthogonal direct sum, 5, 119 overlattice

even, 103

packing radius, 111 parallelotope

fundamental, 1 parity check matrix, 9 Parker, Conway, and Sloane

theorems of, 115 perfect code, 66 Pierce, Gleason, and Sloane

theorem of, 165 plane

unimodular hyperbolic, 119 points of a design, 65 Poisson summation formula, 44 poles, 162 polynomial

generator, 78 harmonic, 30 monic, 77 spherical, 30

polytope Delaunay, 112

positive definite scalar product totally, 167

positive root, 31 primitive sublattice, 5 primitive vector, 119 problem

Index

covering, 112 kissing number, 111 Newton number, 111

product of fractional ideals, 167 puncturing of a code, 67

quadratic form discriminant, 103 finite, 102

quadratic residue, 82 quadratic residue code, 82

radius

rate

covering, 112 packing, 111

information, 7 real

totally, 167 reciprocity law for Gaussian sums, 96 reducible lattice, 22 Reed-Solomon code, 79 reflection, 25 residue

quadratic, 82 residue theorem, 55 Riemann (-function, 50 Riemann-Roch theorem, 55 root, 16

highest, 32 root lattice, 16 RS code, 79

scalar product totally positive definite, 167

self-dual code, 9 semi-direct product, 15 shallow hole, 113 simplex code, 80 simply transitively, 132 Sloane and Conway

theorem of, 118

theorem of Conway, 130 Sloane, Conway, and Parker

theorems of, 115 Sloane, Gleason, and Pierce

theorem of, 165 sphere packing, 111 spherical of degree r, 87 spherical polynomial, 30 split exact sequence, 15 stabilizer, 41, 132 Steiner system, 64, 65 sublattice, 4

primitive, 5 sum

orthogonal direct, 5 summation formula

Poisson, 44 surface

Hilbert modular, 161 symmetric bilinear form, 138 symmetric bilinear form module

unimodular, 4

187

symmetric bilinear form module, 3 automorphism group of, 3 even, 5

symmetric bilinear form modules automorphism of, 3 isomorphic, 3

symmetric Hilbert modular form, 148 symmetry group of a tetrahedron,

151 syndrome, 9 system

Steiner, 64, 65 tetrad, 105

system of roots fundamental, 17

ternary code, 6 ternary codes

MacWilliams identity for, 156 ternary Golay code

188

extended, 145 tetrad code, 105 tetrad system, 105 tetrads, 105 tetrahedral equation, 158 tetrahedron, 151 theorem

Conway's, 109 Gleason's, 75 Klein's, 164 Niemeier's, 106

theorem of Alpbach, 149 theorem of Brom§ and Enguehard,

157 theorem of Conway, 132 theorem of Conway and Sloane, 118,

130 theorem of Gleason, Pierce, and

Sloane, 165 theorem of Hirzebruch, 164 theorem of van der Geer and Hirze­

bruch, 149 theorems of Conway, Parker, and

Sloane, 115 theta function, 39

extremal, 77 totally positive definite scalar prod-

uct, 167 totally real, 167 trace, 136 transform

Fourier, 44 transitive

doubly, 84 transitively, 25

simply, 132 trivial code, 6

unimodular, 4 unimodular hyperbolic plane, 119 unimodular lattice, 4

van der Geer and Hirzebruch theorem of, 149

vector glue, 116 isotropic, 119 primitive, 119 Weyl, 33, 116

volume, 1

weight, 7 modular form of, 43

weight enumerator extremal, 76 Hamming, 62 Lee, 149

Weyl group, 25 Weyl vector, 33, 116 wordlength, 6

Index

Milnor's Textbook on Dynamics

John Milnor Dynamics in One Complex Variable Introductory Lectures 2. ed. 2000. viii, 257 pp. Softc. € 26,00 ISBN 3-528-13130-6 Contents: Chronological Table - Riemann Surfaces - Iterated Holomor­phic Maps - Local Fixed Point Theory - Periodic Points: Global Theory -Structure of the Fatou Set - Using the Fatou Set to study the Julia Set­Appendices

This text studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. The subject is large and rapidly growing. These notes are intended to introduce the reader to some key ideas in the field, and to form a basis for further study. The reader is assumed to be familiar with the rudiments of complex variable theory and of two-dimensional differential geometry, as well as some basic topics from topology.

II vleweg

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Stand 1.4.2002. Anderungen vorbehalten. Erhaltlich im Buchhandel oder im Verlag.

Discrete mathematics in relation to computer science

Hans Jiirgen Pro mel, Angelika Steger The Steiner Tree Problem A Tour through Graphs, Algorithms, and Complexity

2002. viii, 241 pp. Softc. € 29,50 ISBN 3-528-06762-4 Basics: Graphs, Algorithms, Complexity - Special Terminal Sets - Exact Algorithms - Approximation Algorithms - Randomness Helps - Limits of Approximability - Geometric Steiner Problems

In recent years, algorithmic graph theory has become increasingly important as a link between discrete mathematics and theoretical computer science. This textbook introduces students of mathematics and computer science to the interrelated fields of graphs theory, algo­rithms and complexity. No specific previous knowledge is assumed.

The central theme of the book is a geometrical problem dating back to Jakob Steiner. This problem, now called the Steiner problem, was initi­ally of importance only within the context of land surveying. In the last decade, however, applications as diverse as VLSI-Iayout and the study of phylogenetiC trees led to a rapid rise of interest in this pro­blem. The resulting progress has uncovered fascinating connections between and within graph theory, the study of algorithms, and com­plexity theory. This single problem thus serves to bind and motivate these areas. The book's topics include: exact algorithms, computatio­nal complexity, approximation algorithms, the use of randomness, limits of approximability.

A special feature of the book is that each chapter ends with an "excur­sion" into some related area. These excursions reinforce the concepts and methods introduced for the Steiner problem by placing them in a broader context.

aI vleweg

Abraham-Lincoln-StraBe 46 65189 Wiesbaden Fax 0611.7878-400 www.vieweg.de

Stand 1.4.2002. Anderungen vorbehalten. Erhiiltlich im Buchhandel oder im Verlag.