The Theory of Error-Correcting Codes

Volume 16

By: F. J. Macwilliams, N. J. a. Sloane

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Hardcover | 1 June 1988 | Edition Number 11

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This work presents a unified account of all the mathematical techniques used to date. It is presented in an intelligible manner and is designed as both introductory textbook for the beginner and reference book for the expert engineer and mathematician. The book is divided into sections which can be used as a basis for an elementary course on coding theory for mathematicians, a second course for mathematicians, an elementary first course for engineers, and a second course for engineers. The remainder is directed towards the experts in the subject and is suitable for advanced courses and seminars. An abundance of problems is included, as well as a number of unsolved problems suitable for use in research projects as topics for study. Engineering Societies Library This text provides a thorough coverage of mathematical techniques applicable to this subject. The set can serve as a text for advanced students or as a reference work for the practitioner. New Technical Books

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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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The Theory of Error-Correcting Codes

Volume 16

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