| Preface | p. v |
| Preface to the third printing | p. xi |
| Contents | p. xiii |
| Linear codes | |
| Linear codes | p. 1 |
| Properties of a linear code | p. 5 |
| At the receiving end | p. 7 |
| More about decoding a linear code | p. 15 |
| Error probability | p. 18 |
| Shannon's theorem on the existence of good codes | p. 22 |
| Hamming codes | p. 23 |
| The dual code | p. 26 |
| Construction of new codes from old (II) | p. 27 |
| Some general properties of a linear code | p. 32 |
| Summary of Chapter 1 | p. 34 |
| Notes on Chapter 1 | p. 34 |
| Nonlinear codes, Hadamard matrices, designs and the Golay code | |
| Nonlinear codes | p. 38 |
| The Plotkin bound | p. 41 |
| Hadamard matrices and Hadamard codes | p. 44 |
| Conferences matrices | p. 55 |
| t-designs | p. 58 |
| An introduction to the binary Golay code | p. 64 |
| The Steiner system S(5, 6, 12), and nonlinear single-error correcting codes | p. 70 |
| An introduction to the Nordstrom-Robinson code | p. 73 |
| Construction of new codes from old (III) | p. 76 |
| Notes on Chapter 2 | p. 78 |
| An introduction to BCH codes and finite fields | |
| Double-error-correcting BCH codes (I) | p. 80 |
| Construction of the field GF(16) | p. 82 |
| Double-error-correcting BCH codes (II) | p. 86 |
| Computing in a finite field | p. 88 |
| Notes on Chapter 3 | p. 92 |
| Finite fields | |
| Introduction | p. 93 |
| Finite fields: the basic theory | p. 95 |
| Minimal polynomials | p. 99 |
| How to find irreducible polynomials | p. 107 |
| Tables of small fields | p. 109 |
| The automorphism group of GF(p[superscript m]) | p. 112 |
| The number of irreducible polynomials | p. 114 |
| Bases of GF(p[superscript m]) over GF(p) | p. 115 |
| Linearized polynomials and normal bases | p. 118 |
| Notes on Chapter 4 | p. 124 |
| Dual codes and their weight distribution | |
| Introduction | p. 125 |
| Weight distribution of the dual of a binary linear code | p. 125 |
| The group algebra | p. 132 |
| Characters | p. 134 |
| MacWilliams theorem for nonlinear codes | p. 135 |
| Generalized MacWilliams theorems for linear codes | p. 141 |
| Properties of Krawtchouk polynomials | p. 150 |
| Notes on Chapter 5 | p. 153 |
| Codes, designs and perfect codes | |
| Introduction | p. 155 |
| Four fundamental parameters of a code | p. 156 |
| An explicit formula for the weight and distance distribution | p. 158 |
| Designs from codes when s [less than or equal] d' | p. 160 |
| The dual code also gives designs | p. 164 |
| Weight distribution of translates of a code | p. 166 |
| Designs from nonlinear codes when s' < d | p. 174 |
| Perfect codes | p. 175 |
| Codes over GF(q) | p. 176 |
| There are no more perfect codes | p. 179 |
| Notes on Chapter 6 | p. 186 |
| Cyclic codes | |
| Introduction | p. 188 |
| Definition of a cyclic code | p. 188 |
| Generator polynomial | p. 190 |
| The check polynomial | p. 194 |
| Factors of x[superscript n] - 1 | p. 196 |
| t-error-correcting BCH codes | p. 201 |
| Using a matrix over GF(q[superscript n]) to define a code over GF(q) | p. 207 |
| Encoding cyclic codes | p. 209 |
| Notes on Chapter 7 | p. 214 |
| Cyclic codes (contd.): Idempotents and Mattson-Solomon polynomials | |
| Introduction | p. 216 |
| Idempotents | p. 217 |
| Minimal ideals, irreducible codes, and primitive idempotents | p. 219 |
| Weight distribution of minimal codes | p. 227 |
| The automorphism group of a code | p. 229 |
| The Mattson-Solomon polynomial | p. 239 |
| Some weight distributions | p. 251 |
| Notes on Chapter 8 | p. 255 |
| BCH codes | |
| Introduction | p. 257 |
| The true minimum distance of a BCH code | p. 259 |
| The number of information symbols in BCH codes | p. 262 |
| A table of BCH codes | p. 266 |
| Long BCH codes are bad | p. 269 |
| Decoding BCH codes | p. 270 |
| Quadratic equations over GF(2[superscript m]) | p. 277 |
| Double-error-correcting BCH codes are quasi-perfect | p. 279 |
| The Carlitz-Uchiyama bound | p. 280 |
| Some weight distributions are asymptotically normal | p. 282 |
| Notes on Chapter 9 | p. 291 |
| Reed-Solomon and Justesen codes | |
| Introduction | p. 294 |
| Reed-Solomon codes | p. 294 |
| Extended RS codes | p. 296 |
| Idempotents of RS codes | p. 296 |
| Mapping GF(2[superscript m]) codes into binary codes | p. 298 |
| Burst error correction | p. 301 |
| Encoding Reed-Solomon codes | p. 301 |
| Generalized Reed-Solomon codes | p. 303 |
| Redundant residue codes | p. 305 |
| Decoding RS codes | p. 306 |
| Justesen codes and concatenated codes | p. 306 |
| Notes on Chapter 10 | p. 315 |
| MDS codes | |
| Introduction | p. 317 |
| Generator and parity check matrices | p. 318 |
| The weight distribution of an MDS code | p. 319 |
| Matrices with every square submatrix nonsingular | p. 321 |
| MDS codes from RS codes | p. 323 |
| n-arcs | p. 326 |
| The known results | p. 327 |
| Orthogonal arrays | p. 328 |
| Notes on Chapter 11 | p. 329 |
| Alternant, Goppa and other generalized BCH codes | |
| Introduction | p. 332 |
| Alternant codes | p. 333 |
| Goppa codes | p. 338 |
| Further properties of Goppa codes | p. 346 |
| Extended double-error-correcting Goppa codes are cyclic | p. 350 |
| Generalized Srivastava codes | p. 357 |
| Chien-Choy generalized BCH codes | p. 360 |
| The Euclidean algorithm | p. 362 |
| Decoding alternant codes | p. 365 |
| Notes on Chapter 12 | p. 368 |
| Reed-Muller codes | |
| Introduction | p. 370 |
| Boolean functions | p. 370 |
| Reed-Muller Codes | p. 373 |
| RM codes and geometries | p. 377 |
| The minimum weight vectors generate the code | p. 381 |
| Encoding and decoding (I) | p. 385 |
| Encoding and decoding (II) | p. 388 |
| Other geometrical codes | p. 397 |
| Automorphism groups of the RM codes | p. 398 |
| Mattson-Solomon polynomials of RM codes | p. 401 |
| The action of the general affine group on Mattson-Solomon polynomials | p. 402 |
| Notes on Chapter 13 | p. 403 |
| First-order Reed-Muller codes | |
| Introduction | p. 406 |
| Pseudo-noise sequences | p. 406 |
| Cosets of the first-order Reed-Muller code | p. 412 |
| Encoding and decoding R(1, m) | p. 419 |
| Bent functions | p. 426 |
| Notes on Chapter 14 | p. 431 |
| Second-order Reed-Muller, Kerdock and Preparata codes | |
| Introduction | p. 433 |
| Weight distribution of second-order Reed-Muller codes | p. 434 |
| Weight distribution of arbitrary Reed-Muller codes | p. 445 |
| Subcodes of dimension 2m of R(2, m)* and R(2, m) | p. 448 |
| The Kerdock code and generalizations | p. 453 |
| The Preparata code | p. 466 |
| Goethals' generalization of the Preparata codes | p. 476 |
| Notes on Chapter 15 | p. 477 |
| Quadratic-residue codes | |
| Introduction | p. 480 |
| Definition of quadratic-residue codes | p. 481 |
| Idempotents of quadratic-residue codes | p. 484 |
| Extended quadratic-residue codes | p. 488 |
| The automorphism group of QR codes | p. 491 |
| Binary quadratic residue codes | p. 494 |
| Double circulant and quasi-cyclic codes | p. 505 |
| Quadratic-residue and symmetry codes over GF(3) | p. 510 |
| Decoding of cyclic codes and others | p. 512 |
| Notes on Chapter 16 | p. 518 |
| Bounds on the size of a code | |
| Introduction | p. 523 |
| Bounds on A(n, d, w) | p. 524 |
| Bounds on A(n, d) | p. 531 |
| Linear programming bounds | p. 535 |
| The Griesmer bound | p. 546 |
| Constructing linear codes; anticodes | p. 547 |
| Asymptotic bounds | p. 556 |
| Notes on Chapter 17 | p. 566 |
| Methods for combining codes | |
| Introduction | p. 567 |
| Product codes and generalizations | |
| Direct product codes | p. 568 |
| Not all cyclic codes are direct products of cyclic codes | p. 571 |
| Another way of factoring irreducible cyclic codes | p. 573 |
| Concatenated codes: the * construction | p. 575 |
| A general decomposition for cyclic codes | p. 578 |
| Other methods of combining codes | |
| Methods which increase the length | p. 581 |
| Construction X: adding tails to the codewords | p. 581 |
| Construction X4: combining four codes | p. 584 |
| Single- and double-error-correcting codes | p. 586 |
| The [vertical bar] a + x [vertical bar] b + x [vertical bar] a + b + x [vertical bar] construction | p. 587 |
| Piret's construction | p. 588 |
| Constructions related to concatenated codes | p. 589 |
| A method for improving concatenated codes | p. 589 |
| Zinov'ev's generalized concatenated codes | p. 590 |
| Methods for shortening a code | p. 592 |
| Constructions Y1-Y4 | p. 592 |
| A construction of Helgert and Stinaff | p. 593 |
| Notes on Chapter 18 | p. 594 |
| Self-dual codes and invariant theory | |
| Introduction | p. 596 |
| An introduction to invariant theory | p. 598 |
| The basic theorems of invariant theory | p. 607 |
| Generalizations of Gleason's theorems | p. 617 |
| The nonexistence of certain very good codes | p. 624 |
| Good self-dual codes exist | p. 629 |
| Notes on Chapter 19 | p. 633 |
| The Golay codes | |
| Introduction | p. 634 |
| The Mathieu group M[subscript 24] | p. 636 |
| M[subscript 24] is five-fold transitive | p. 637 |
| The order of M[subscript 24] is 24.23.22.21.20.48 | p. 638 |
| The Steiner system S(5, 8, 24) is unique | p. 641 |
| The Golay codes g[subscript 23] and g[subscript 24] are unique | p. 646 |
| The automorphism groups of the ternary Golay codes | p. 647 |
| The Golay codes g[subscript 11] and g[subscript 12] are unique | p. 648 |
| Notes on Chapter 20 | p. 649 |
| Association schemes | |
| Introduction | p. 651 |
| Association schemes | p. 651 |
| The Hamming association scheme | p. 656 |
| Metric schemes | p. 659 |
| Symplectic forms | p. 661 |
| The Johnson scheme | p. 665 |
| Subsets of association schemes | p. 666 |
| Subsets of symplectic forms | p. 667 |
| t-designs and orthogonal arrays | p. 670 |
| Notes on Chapter 21 | p. 671 |
| Tables of the best codes known | |
| Introduction | p. 673 |
| Figure 1, a small table of A(n, d) | p. 683 |
| Figure 2, an extended table of the best codes known | p. 690 |
| Figure 3, a table of A(n, d, w) | p. 691 |
| Finite geometries | |
| Introduction | p. 692 |
| Finite geometries, PG(m, q) and EG(m, q) | p. 692 |
| Properties of PG(m, q) and EG(m, q) | p. 697 |
| Projective and affine planes | p. 701 |
| Notes on Appendix B | p. 702 |
| Bibliography | p. 703 |
| Index | p. 757 |
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