Book by J. Seberry: Orthogonal Designs – Hadamard Matrices, Quadratic Forms and Algebras


Communicated by Nek Valous, National Center for Tumor Diseases (NCT), Heidelberg.

Jennifer Seberry, Orthogonal Designs – Hadamard Matrices, Quadratic Forms and Algebras, Springer, Berlin, 2017, Hardcover 117,69 €, URL: http://www.springer.com/de/book/9783319590318

Description (publisher): 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.

Contents
1 Orthogonal Designs . . . . . . . . . . . . . . 1
1.1 Hurwitz-Radon families . . . . . . . . . . . . . . 2
2 Non-existence Results . . . . . . . . . . . . . . . . . 7
2.1 Weighing Matrices . . . . . . . . . . . . . . . 7
2.2 Odd Order . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Algebraic Problem . . . . . . . . . . . . . . . . . 13
2.4 Orthogonal Designs’ Algebraic Problem . . . . . . . . . . . . . . . . . . . 13
2.5 Geramita-Verner Theorem Consequences . . . . . . . . . . . . . . . . . . 15
3 Algebraic Theory of Orthogonal Designs . . . . . . . . . . . . . . . . . . 19
3.1 Generalities on Quadratic and Bilinear Forms . . . . . . . . . . . . . . 19
3.2 The Matrix Formulation . . . . . . . . . . . . . 21
3.3 Mapping Between Bilinear Spaces . . . . . . 22
3.4 New Spaces From Old. . . . . . . . . . . . . . . . 23
3.5 Bilinear Spaces Classification Theorems . . . . . . . . . . . . . . . . . . . 24
3.6 Classification of Quadratic Forms Over Q. . . . . . . . . . . . . . . . . . 25
3.7 The Similarities of a Bilinear Space . . . . . . . . . . 30
3.8 Linear Subspaces of Sim(V ) . . . . . . . . . . . . . . . 31
3.9 Relations Between Rational Families in the Same Order . . . . . 36
3.10 Clifford Algebras . . . . . . . . . .. . . . . . . . . . . . . . . . . 37
3.11 Similarity Representations . . . . . . . . . . . . . . . . 38
3.12 Some Facts About Positive Definite Forms Over Q. . . . . . . . . . 40
3.13 Reduction to Powers of 2 . . . . . . . . . . . . . . . . . . 43
3.14 Orders 4 and 8 . . . . . . . . . . . . . . . . . . . 46
3.14.1 Order 4 . . . . . . . . . . . . . . . . . . . . . . . 46
3.14.2 Order 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.15 Order 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.15.1 Case 1: 9-member rational families. . . . . . . . . . . . . . . . . . 53
3.15.2 Case 2: 7-member rational families. . . . . . . . . . . . . . . . . . 53
3.15.3 Case 3: 8-member rational families. . . . . . . . . . . . . . . . . . 54
3.16 Order 32 . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.17 Solution of the Algebraic Problem . . . . . . . . . 57
3.18 Combining Algebra with Combinatorics . . . . . . . . . . . . . . . . . . . 59
3.18.1 Alert . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4 Orthogonal Designs Constructed via Plug-in Matrices . . . . 63
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Some Orthogonal Designs Exist . . . . . . . . . . 63
4.3 Some Basic Matrix Results . . . .. . . . . . . . . 68
4.3.1 Supplementary Difference Sets, their Incidence Matrices and their Uses as Suitable Matrices . . . . . . . . . 74
4.4 Existence of Weighing Matrices . . . . . .. . 76
4.5 Constructions for W(h,h) and W(h,h−1) . .. . . . . . . . . . . . . 82
4.6 Using Circulants–Goethals-Seidel Array and Kharaghani Array 89
4.7 Constraints on construction using circulant matrices . . . . . . . . 95
4.8 Eades’ Technique for Constructing Orthogonal Designs . . . . . . 96
4.9 Some Arrays for Eight Circulants . . . . . . . . . . . . 107
4.10 Amicable Sets and Kharaghani Arrays . . . . . . . . . . . 110
4.11 Construction using 8 Disjoint Matrices . . .. . . . 111
4.11.1 Hadamard Matrices . . . . . . . . . . . . 115
4.12 Baumert-Hall Arrays . . . . . . . . . . . . . . . . 117
4.13 Plotkin Arrays . . . . . . . . . . . . . . . . . . . . . . . . 124
4.13.1 Kharaghani’s Plotkin arrays . . . . . . . . . . . . 126
4.14 More Specific Constructions using Circulant Matrices . . . . . . . 126
4.15 Generalized Goethals-Seidel Arrays . . . . . .. . . . 129
4.15.1 Some Infinite Families of Orthogonal Designs . . . . . . . . 133
4.15.2 Limitations . . . . . . . . . . . . . . . . . . . . . . 134
4.16 Balanced Weighing Matrices . . . .. . . . . . . . . 134
4.16.1 Necessary Conditions for the Existence of Balanced Weighing Matrices . .  . . 135
4.16.2 Construction Method for Balanced Weighing Designs . 136
4.16.3 Regular Balanced Weighing Matrices . . . . . . . . . . . 139
4.16.4 Application of the Frobenius Group Determinant
Theorem to Balanced Weighing Matrices . . . . . . . . . . . . 141
4.16.5 Balanced Weighing Matrices with v ≤ 25 . . .. . . . . 143
4.16.6 No Circulant Balanced Weighing Matrices BW(v,v−1) Based on (v,v−1,v−2) Configurations . 144
4.17 Negacyclic Matrices . . . . . . . . . . . . . . . . . . 148
4.17.1 Constructions . . . . . . . .. . . . . . . . . . . 152
4.17.2 Applications . . . . . . . . . . . . . . . . . . . . 153
4.17.3 Combinatorial Applications . . . . . . . 154
5 Amicable Orthogonal Designs. .. . . . . . . . . . . . 155
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . 155
5.2 Definitions and Elementary Observations . . . . . . . . . . . . . . . . . . 157
5.2.1 n Odd . . . . . . . . . . . . . . . . . . . . . . . 158
5.2.2 n = 2b, b Odd . . . . . . . . . . . . . . . 160
5.3 More on Variables in an Amicable Orthogonal Design . . . . . . . 162
5.4 The Number of Variables . . . . . . . . . . . . . . . . 164
5.5 The Algebraic Theory of Amicable Orthogonal Designs . . . . . . 168
5.6 The Combinatorial Theory of Amicable Orthogonal Designs . 171
5.6.1 Cases a = 2,3 or 4 . . . . . . . . . . . . . . . . 175
5.7 Construction of Amicable Orthogonal Designs . . . . . . . . . . . . . . 178
5.8 Construction Methods . . . . . . . . .. . . . . . . . . 182
5.9 Specific Orders 2n . . . . . . . . . . . . . . . . . . . 183
5.9.1 Amicable OD of order 2 . . . . . . . . . . . . 183
5.9.2 Amicable Orthogonal Designs of Order 8 . . . . . . . . . . . . 184
5.10 Amicable Hadamard Matrices . . . . . . . . . 194
5.11 Amicable Hadamard Matrices and Cores . . . . . . . . . 202
5.12 Strong Amicable Designs . . . . . .. . . . . . . . . 205
5.13 Structure of Amicable Weighing Matrices . . . . .. . . . . . . . . 206
5.14 Generalizations . . . . . . . . . . . . . . . . . . . . . . . 207
5.15 Repeat and Product Design Families . . . . . . . . . 211
6 Product Designs and Repeat Designs (Gastineau-Hills) . . . 213
6.1 Generalizing Amicable Orthogonal Designs . . . . . . . . . . . . . . . . 213
6.1.1 Product Designs . . . . . . . . . . . . . . . . . . . . . 214
6.1.2 Constructing Product Designs . . . . . . . . . . . . . . . . . . . . . 215
6.2 Constructing Orthogonal Designs from Product Designs . . . . . 218
6.2.1 Applications . . . . . . . . . . . . . . . . . . . . 221
6.3 Using Families of Matrices – Repeat Designs . . . .. . . 221
6.3.1 Construction and Replication of Repeat Designs . . . . . . 224
6.3.2 Construction of Orthogonal Designs . . . . . . . . . . . . . . . 225
6.4 Gastineau-Hills on Product Designs and Repeat Designs . . . . . 227
6.5 Gastineau-Hills Systems of Orthogonal Designs . . . . . . . . . . . . . 232
6.6 Clifford-Gastineau-Hills Algebras . . . . . . . . . . . . 236
6.7 Decomposition . . . . . . . . . . . . . . . . . . . . . . 238
6.8 Clifford-Gastineau-Hills (CGH) Quasi Clifford Algebras . . . . . 242
6.9 The Order Number Theorem . . . . . . . . . . . . . . . 246
6.10 Periodicity. . . . . . . . . . . . . . . . . . . . . . . . . . 253
6.11 Orders of Repeat Designs . . . . . . . . . . . . . 256
6.12 Orders of Product Designs and Amicable Sets . . . . . . . . . . . . . . 261
7 Techniques . . . . . . . . . . . . .. . . . . . . . . . . . . . 267
7.1 Using Cyclotomy . . . . . . . .. . . . . . . . . . . . . . 267
7.2 Sequences with Zero-autocorrelation Function . . . . . . . 275
7.2.1 Other sequences with zero auto-correlation function . . . 282
7.3 Current Results for Non-Periodic Golay Pairs . . .. . . . . 284
7.4 Recent Results for Periodic Golay Pairs . . . 285
7.5 Using complementary sequences to form Baumert-Hall arrays 285
7.6 Construction using complementary sequences . . . . . . 291
7.7 6-Turyn-type Sequences . . . .. . . . . . . . . . . . . . . 294
8 Robinson’s Theorem . . . . . . . . . . . . . . . . . . . . 295
9 Hadamard Matrices and Asymptotic Orthogonal Designs …. 305
9.1 Existence of Hadamard Matrices . . . . . . . . . . . 305
9.2 The Existence of Hadamard Matrices . . . . . . . 306
9.3 Asymptotic Existence Results for Orthogonal Designs . . . . . . . 309
9.4 n-Tuples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
9.4.1 Description of the Construction Algorithm . . . . . . . . 316
9.4.2 Implementing the Algorithm . . . . . . . . . . . . . . . . 318
9.4.3 n-Tuples in Powers of 2 With No Zeros . . . . . . . . . . 319
9.5 Enough Powers of Two: Asymptotic Existence . . . . . . . . . . 321
9.5.1 The Asymptotic Hadamard Existence Theorem . . . . . . 323
9.5.2 Ghaderpour and Kharaghani’s Uber Asymptotic Results323
9.6 The Asymptotic Existence of Amicable Orthogonal Designs . . 329
9.7 de Launey’s Theorem . .. . . . . . . . . . 332
10 Complex, Quaternion and Non Square Orthogonal Designs … 335
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 335
10.2 Complex orthogonal designs . . . . . . . . . . . . . . 336
10.3 Amicable orthogonal designs of quaternions . . . . . . . . . . . 337
10.4 Construction techniques . . . . . . . . . . . . . . . . . . . 340
10.4.1 Amicable orthogonal designs . . . . .. . . . . . . . . 341
10.5 Amicable orthogonal design of quaternions . .. . . . 342
10.6 Combined Quaternion Orthogonal Designs from Amicable Designs . . .  . 348
10.7 Le Tran’s Complex Orthogonal Designs of Order Eight . . . . . . 352
10.8 Research Problem . . . . . . . . . . . . . . . . . . . . . . . . 355
A Orthogonal Designs in Order 12,24,48 and 3.q . . . . . . . . . . 357
A.1 Number of possible n-tuples . . . . . . . . . . . . . 357
A.2 Some Theorems . . . . . . . . . . . . . . . . . . . 358
A.3 Order 12 . . . . . . . . . . . . . . . . . . . . . . . . . 358
A.4 Order 24 . . . .. . . . . . . . . . . . . . . . . . . . 360
A.5 Order 48 . . . . . . . . .. . . . . . . . . . . . . . . . . . 366
B Orthogonal Designs in Order 20, 40 and 80 . . . . . . . . . . . . . . . 369
B.1 Some Theorems . . . . . . . . . . . . . . . . . . . . . . . . 369
B.2 Order 20 . . . . . . . . . . . . . . . . . . . . . . . . 369
B.3 Order 40 . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
B.4 Order 80 . . .. . . . . . . . . . . . . . . . .. . . 375
C Orthogonal Designs in Order 28 and 56 . . . . . 379
C.1 Some Theorems . . . . . . . . . . . . . . . 379
C.2 Order 28 .. . . . . . . . . . . . . . . . 379
C.3 Order 56 . . . . . . . . . . . . . . . 385
C.4 Further Research . .. . . . . . . . . . . 385
D Orthogonal Designs in Order 36 and 72 . . . . . . . . . . . . 389
D.1 Some theorems . . . . . . . . . . 389
D.2 Order 36 . . .. . . . . . . . . . . . . . 389
D.3 Order 72 . .. . . . . . . . . . . . . . . 390
E Orthogonal Designs in order 44 . . . . . . . . . 395
E.1 Some theorems . . . . . . . . . . . . . . 395
E.2 Order 44 .. . . . . . . . . . . . . . 395
F Orthogonal Designs in Powers of 2 . . . . 403
F.1 Some Theorems . . . . . . . . . . 403
F.2 Orthogonal Designs in Order 16 . . . . . . 404
F.3 Order 32 . . . 409
F.4 Order 64 . .. . . . . . . . . . 415
G Some Complementary Sequences . . . .. . . . . . . . . 417
H Product Designs . . . . . . . . .. . . . . . . . . 425
References . . . . . . . . .429
Index . . . . . . . . . . . . . . . 441

Source: Email from Nek Valous, nek.valous_AT_nct-heidelberg.de, 10/02/2018 05:26, http://www.springer.com/de/book/9783319590318, http://www.springer.com/cda/content/document/cda_downloaddocument/9783319590318-t1.pdf?SGWID=0-0-45-1626527-p180864462, accessed: 12 Feb. 2018

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