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E-raamat: Orthogonal Designs: Hadamard Matrices, Quadratic Forms and Algebras

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  • Formaat: PDF+DRM
  • Ilmumisaeg: 20-Nov-2017
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319590325
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Orthogonal Designs: Hadamard Matrices, Quadratic Forms and AlgebrasSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 20-Nov-2017
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319590325
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Orthogonal designs have proved fundamental to constructing code division multiple antenna systems for more efficient mobile communications. Starting with basic theory, this book develops the algebra and combinatorics to create new communications modes. Intended primarily for researchers, it is also useful for graduate students wanting to understand some of the current communications coding theories.

1 Orthogonal Designs
1(6)
1.1 Hurwitz-Radon families
2(5)
2 Non-existence Results
7(12)
2.1 Weighing Matrices
7(1)
2.2 Odd Order
8(5)
2.3 Algebraic Problem
13(1)
2.4 Orthogonal Designs' Algebraic Problem
13(2)
2.5 Geramita-Verner Theorem Consequences
15(4)
3 Algebraic Theory of Orthogonal Designs
19(44)
3.1 Generalities on Quadratic and Bilinear Forms
19(2)
3.2 The Matrix Formulation
21(1)
3.3 Mapping Between Bilinear Spaces
22(1)
3.4 New Spaces From Old
23(1)
3.5 Bilinear Spaces Classification Theorems
24(1)
3.6 Classification of Quadratic Forms Over Q
25(5)
3.7 The Similarities of a Bilinear Space
30(1)
3.8 Linear Subspaces of Sim(V)
31(5)
3.9 Relations Between Rational Families in the Same Order
36(1)
3.10 Clifford Algebras
37(1)
3.11 Similarity Representations
38(2)
3.12 Some Facts About Positive Definite Forms Over Q
40(3)
3.13 Reduction to Powers of 2
43(3)
3.14 Orders 4 and 8
46(5)
3.14.1 Order 4
46(2)
3.14.2 Order 8
48(3)
3.15 Order 16
51(5)
3.15.1 Case 1: 9-member rational families
53(1)
3.15.2 Case 2: 7-member rational families
53(1)
3.15.3 Case 3: 8-member rational families
54(2)
3.16 Order 32
56(1)
3.17 Solution of the Algebraic Problem
57(2)
3.18 Combining Algebra with Combinatorics
59(4)
3.18.1 Alert
61(2)
4 Orthogonal Designs Constructed via Plug-in Matrices
63(92)
4.1 Introduction
63(1)
4.2 Some Orthogonal Designs Exist
63(5)
4.3 Some Basic Matrix Results
68(8)
4.3.1 Supplementary Difference Sets, their Incidence Matrices and their Uses as Suitable Matrices
74(2)
4.4 Existence of Weighing Matrices
76(6)
4.5 Constructions for W(h, h) and W(h, h -- 1)
82(7)
4.6 Using Circulants--Goethals-Seidel Array and Kharaghani Array
89(6)
4.7 Constraints on construction using circulant matrices
95(1)
4.8 Eades' Technique for Constructing Orthogonal Designs
96(11)
4.9 Some Arrays for Eight Circulants
107(3)
4.10 Amicable Sets and Kharaghani Arrays
110(1)
4.11 Construction using 8 Disjoint Matrices
111(6)
4.11.1 Hadamard Matrices
115(2)
4.12 Baumert-Hall Arrays
117(7)
4.13 Plotkin Arrays
124(2)
4.13.1 Kharaghani's Plotkin arrays
126(1)
4.14 More Specific Constructions using Circulant Matrices
126(3)
4.15 Generalized Goethals-Seidel Arrays
129(5)
4.15.1 Some Infinite Families of Orthogonal Designs
133(1)
4.15.2 Limitations
134(1)
4.16 Balanced Weighing Matrices
134(14)
4.16.1 Necessary Conditions for the Existence of Balanced Weighing Matrices
135(1)
4.16.2 Construction Method for Balanced Weighing Designs
136(3)
4.16.3 Regular Balanced Weighing Matrices
139(2)
4.16.4 Application of the Frobenius Group Determinant Theorem to Balanced Weighing Matrices
141(2)
4.16.5 Balanced Weighing Matrices with v ≤ 25
143(1)
4.16.6 No Circulant Balanced Weighing Matrices BW(v, v -- l) Based on (v, v -- 1, v -- 2) Configurations
144(4)
4.17 Negacyclic Matrices
148(7)
4.17.1 Constructions
152(1)
4.17.2 Applications
153(1)
4.17.3 Combinatorial Applications
154(1)
5 Amicable Orthogonal Designs
155(58)
5.1 Introduction
155(2)
5.2 Definitions and Elementary Observations
157(5)
5.2.1 n Odd
158(2)
5.2.2 n = 2b, b Odd
160(2)
5.3 More on Variables in an Amicable Orthogonal Design
162(2)
5.4 The Number of Variables
164(4)
5.5 The Algebraic Theory of Amicable Orthogonal Designs
168(3)
5.6 The Combinatorial Theory of Amicable Orthogonal Designs
171(7)
5.6.1 Cases a = 2, 3 or 4
175(3)
5.7 Construction of Amicable Orthogonal Designs
178(4)
5.8 Construction Methods
182(1)
5.9 Specific Orders 2n
183(11)
5.9.1 Amicable OD of order 2
183(1)
5.9.2 Amicable Orthogonal Designs of Order 8
184(10)
5.10 Amicable Hadamard Matrices
194(8)
5.11 Amicable Hadamard Matrices and Cores
202(3)
5.12 Strong Amicable Designs
205(1)
5.13 Structure of Amicable Weighing Matrices
206(1)
5.14 Generalizations
207(4)
5.15 Repeat and Product Design Families
211(2)
6 Product Designs and Repeat Designs (Gastineau-Hills)
213(54)
6.1 Generalizing Amicable Orthogonal Designs
213(5)
6.1.1 Product Designs
214(1)
6.1.2 Constructing Product Designs
215(3)
6.2 Constructing Orthogonal Designs from Product Designs
218(3)
6.2.1 Applications
221(1)
6.3 Using Families of Matrices -- Repeat Designs
221(6)
6.3.1 Construction and Replication of Repeat Designs
224(1)
6.3.2 Construction of Orthogonal Designs
225(2)
6.4 Gastineau-Hills on Product Designs and Repeat Designs
227(5)
6.5 Gastineau-Hills Systems of Orthogonal Designs
232(4)
6.6 Clifford-Gastineau-Hills Algebras
236(2)
6.7 Decomposition
238(4)
6.8 Clifford-Gastineau-Hills (CGH) Quasi Clifford Algebras
242(4)
6.9 The Order Number Theorem
246(7)
6.10 Periodicity
253(3)
6.11 Orders of Repeat Designs
256(5)
6.12 Orders of Product Designs and Amicable Sets
261(6)
7 Techniques
267(28)
7.1 Using Cyclotomy
267(8)
7.2 Sequences with Zero-autocorrelation Function
275(9)
7.2.1 Other sequences with zero auto-correlation function
282(2)
7.3 Current Results for Non-Periodic Golay Pairs
284(1)
7.4 Recent Results for Periodic Golay Pairs
285(1)
7.5 Using complementary sequences to form Baumert-Hall arrays
285(6)
7.6 Construction using complementary sequences
291(3)
7.7 6-Turyn-type Sequences
294(1)
8 Robinson's Theorem
295(10)
9 Hadamard Matrices and Asymptotic Orthogonal Designs
305(30)
9.1 Existence of Hadamard Matrices
305(1)
9.2 The Existence of Hadamard Matrices
306(3)
9.3 Asymptotic Existence Results for Orthogonal Designs
309(5)
9.4 n-Tuples
314(7)
9.4.1 Description of the Construction Algorithm
316(2)
9.4.2 Implementing the Algorithm
318(1)
9.4.3 n-Tuples in Powers of 2 With No Zeros
319(2)
9.5 Enough Powers of Two: Asymptotic Existence
321(8)
9.5.1 The Asymptotic Hadamard Existence Theorem
323(1)
9.5.2 Ghaderpour and Kharaghani's Uber Asymptotic Results
323(6)
9.6 The Asymptotic Existence of Amicable Orthogonal Designs
329(3)
9.7 de Launey's Theorem
332(3)
10 Complex, Quaternion and Non Square Orthogonal Designs
335(22)
10.1 Introduction
335(1)
10.2 Complex orthogonal designs
336(1)
10.3 Amicable orthogonal designs of quaternions
337(3)
10.4 Construction techniques
340(2)
10.4.1 Amicable orthogonal designs
341(1)
10.5 Amicable orthogonal design of quaternions
342(6)
10.6 Combined Quaternion Orthogonal Designs from Amicable Designs
348(4)
10.7 Le Tran's Complex Orthogonal Designs of Order Eight
352(3)
10.8 Research Problem
355(2)
A Orthogonal Designs in Order 12, 24, 48 and 3.q
357(12)
A.1 Number of possible n-tuples
357(1)
A.2 Some Theorems
358(1)
A.3 Order 12
358(2)
A.4 Order 24
360(6)
A.5 Order 48
366(3)
B Orthogonal Designs in Order 20, 40 and 80
369(10)
B.1 Some Theorems
369(1)
B.2 Order 20
369(1)
B.3 Order 40
370(5)
B.4 Order 80
375(4)
C Orthogonal Designs in Order 28 and 56
379(10)
C.1 Some Theorems
379(1)
C.2 Order 28
379(6)
C.3 Order 56
385(1)
C.4 Further Research
385(4)
D Orthogonal Designs in Order 36 and 72
389(6)
D.1 Some theorems
389(1)
D.2 Order 36
389(1)
D.3 Order 72
390(5)
E Orthogonal Designs in order 44
395(8)
E.1 Some theorems
395(1)
E.2 Order 44
395(8)
F Orthogonal Designs in Powers of 2
403(14)
F.1 Some Theorems
403(1)
F.2 Orthogonal Designs in Order 16
404(5)
F.3 Order 32
409(6)
F.4 Order 64
415(2)
G Some Complementary Sequences
417(8)
H Product Designs
425(4)
References 429(12)
Index 441
Emeritus Professor Jennifer Seberry is an Australian cryptographer, mathematician, and computer scientist, now at the University of Wollongong. A graduate of UNSW and LaTrobe, she has taught at the Universities of Newcastle, Sydney, UNSW (ADFA) and Wollongong. Her areas of research include discrete and combinatorial mathematics, Hadamard matrices and bent functions for cryptology, and orthogonal designs. She has published over 400 papers and 7 books. She was the first person in Australia to teach computer security to university students. She is highly respected and has been made a Fellow of the International Association for Cryptologic Research and a Chartered Mathematician by the Institute of Mathematics and its Applications.
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