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E-raamat: Introduction to Coding Theory

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(Michigan Technological University, Houghton, USA)
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This book is designed to be usable as a textbook for an undergraduate course or for an advanced graduate course in coding theory as well as a reference for researchers in discrete mathematics, engineering and theoretical computer science. This second edition has three parts: an elementary introduction to coding, theory and applications of codes, and algebraic curves. The latter part presents a brief introduction to the theory of algebraic curves and its most important applications to coding theory.

Preface xxiii
Acknowledgments xxiv
About the author xxvi
I An elementary introduction to coding
1(132)
1 The concept of coding
3(20)
1.1 Bitstrings and binary operations
3(4)
1.2 The Hamming distance
7(2)
1.3 Binary codes
9(3)
1.4 Error-correcting codes in general
12(2)
1.5 The binary symmetric channel
14(5)
1.6 The sphere-packing bound
19(4)
2 Binary linear codes
23(18)
2.1 The concept of binary linear codes
23(4)
2.2 Block coding
27(2)
2.3 The effect of coding
29(1)
2.4 Duality
30(3)
2.5 Binary Hamming and Simplex codes
33(4)
2.6 Principle of duality
37(4)
3 General linear codes
41(30)
3.1 Prime fields
41(2)
3.2 Finite fields
43(4)
3.3 Linear codes over finite fields
47(3)
3.4 Duality and orthogonal arrays
50(8)
3.5 Weight distribution
58(5)
3.6 The game of SET
63(3)
3.7 Syndrome decoding
66(5)
4 Singleton bound and Reed-Solomon codes
71(10)
5 Recursive constructions I
81(12)
5.1 Shortening and puncturing
81(6)
5.2 Concatenation
87(6)
6 Universal hashing
93(4)
7 Designs and the binary Golay code
97(4)
8 Shannon entropy
101(12)
9 Asymptotic results
113(12)
10 Three-dimensional codes, projective planes
125(6)
11 Summary and outlook
131(2)
II Theory and applications of codes
133(262)
12 Subfield codes and trace codes
135(16)
12.1 The trace
135(5)
12.2 Trace codes and subfield codes
140(3)
12.3 Galois closed codes
143(3)
12.4 Automorphism groups
146(5)
13 Cyclic codes
151(38)
13.1 Some primitive cyclic codes of length 15
151(3)
13.2 Theory of cyclic codes
154(16)
13.3 Decoding BCH codes
170(12)
13.4 Constacyclic codes
182(4)
13.5 Remarks
186(3)
14 Recursive constructions, covering radius
189(16)
14.1 Construction X
189(9)
14.2 Covering radius
198(7)
15 The linear programming method
205(34)
15.1 Introduction to linear programming
205(16)
15.2 The Fourier transform
221(9)
15.3 Some explicit LP bounds
230(2)
15.4 The bound of four
232(7)
16 OA in statistics and computer science
239(46)
16.1 OA and independent random variables
239(3)
16.2 Linear shift register sequences
242(8)
16.3 Cryptography and S boxes
250(4)
16.4 Two-point-based sampling
254(2)
16.5 Resilient functions
256(9)
16.6 Derandomization of algorithms
265(5)
16.7 Authentication and universal hashing
270(15)
17 The geometric description of linear codes
285(64)
17.1 Linear codes as sets of points
285(27)
17.2 Quadratic forms, bilinear forms and caps
312(20)
17.3 Caps: Constructions and bounds
332(17)
18 Additive codes and network codes
349(46)
18.1 Basic constructions and applications
349(11)
18.2 The cyclic theory of additive codes
360(13)
18.2.1 Code equivalence and cyclicity
360(10)
18.2.2 The linear case m = 1
370(3)
18.3 Additive quaternary codes: The geometric approach
373(7)
18.4 Quantum codes
380(9)
18.5 Network codes and subspace codes
389(6)
III Codes and algebraic curves
395(92)
19 Introduction
397(8)
19.1 Polynomial equations and function fields
397(4)
19.2 Places of the rational function field
401(4)
20 Function fields, their places and valuations
405(16)
20.1 General facts
405(4)
20.2 Divisors and the genus
409(5)
20.3 The Riemann-Roch theorem
414(3)
20.4 Some hyperelliptic equations
417(4)
21 Determining the genus
421(10)
21.1 Algebraic extensions of function fields
421(2)
21.2 The hyperelliptic case
423(1)
21.3 The Kloosterman codes and curves
424(2)
21.4 Subrings and integrality
426(1)
21.5 The Riemann-Hurwitz formula
427(4)
22 AG codes, Weierstra B points and universal hashing
431(16)
22.1 The basic construction
431(1)
22.2 Pole orders
432(1)
22.3 Examples of function fields and projective equations
433(7)
22.4 The automorphism group
440(3)
22.5 AG codes and universal hashing
443(1)
22.6 The Hasse-Weil bound
444(3)
23 The last chapter
447(40)
23.1 List decoding
447(3)
23.2 Expander codes
450(2)
23.3 tms-nets
452(4)
23.4 Sphere packings
456(10)
23.5 Permutation codes
466(2)
23.6 Designs
468(2)
23.7 Nonlinear codes
470(9)
23.8 Some highly symmetric codes
479(3)
23.9 Small fields
482(1)
23.10 Short codes
483(4)
References 487(16)
Index 503
Jurgen Bierbrauer
Lisainfo e-raamatute kohta Lisainfo e-raamatute kohta
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