viii Contents 13.2. Incidence Matrices for 2-Designs 189 13.3. Finite Projective Planes 191 13.4. Quadratic Forms and the Witt Cancellation Theorem 198 13.5. The Bruck–Ryser–Chowla Theorem 202 Exercises 205 Chapter 14. Hadamard Matrices 207 14.1. Introduction 207 14.2. The Quadratic Residue Matrix and Paley’s Theorem 208 14.3. Results of Williamson 212 14.4. Hadamard Matrices and Block Designs 216 14.5. A Determinant Inequality, Revisited 219 Exercises 219 Chapter 15. Graphs 221 15.1. Definitions 221 15.2. Graphs and Matrices 223 15.3. Walks and Cycles 224 15.4. Graphs and Eigenvalues 226 15.5. Strongly Regular Graphs 227 Exercises 232 Chapter 16. Directed Graphs 235 16.1. Definitions 235 16.2. Irreducibility and Strong Connectivity 238 16.3. Index of Imprimitivity 242 16.4. Primitive Graphs 245 Exercises 247 Chapter 17. Nonnegative Matrices 249 17.1. Introduction 249 17.2. Preliminaries 250 17.3. Proof of Perron’s Theorem 254 17.4. Nonnegative Matrices 258 17.5. Irreducible Matrices 259 17.6. Primitive and Imprimitive Matrices 260 Exercises 262 Chapter 18. Error-Correcting Codes 265 18.1. Introduction 265 18.2. The Hamming Code 266 18.3. Linear Codes: Parity Check and Generator Matrices 267
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