Introduction to Coding TheorySpringer Science & Business Media, 15.12.1998 - 234 sivua It is gratifying that this textbook is still sufficiently popular to warrant a third edition. I have used the opportunity to improve and enlarge the book. When the second edition was prepared, only two pages on algebraic geometry codes were added. These have now been removed and replaced by a relatively long chapter on this subject. Although it is still only an introduction, the chapter requires more mathematical background of the reader than the remainder of this book. One of the very interesting recent developments concerns binary codes defined by using codes over the alphabet 7l.4• There is so much interest in this area that a chapter on the essentials was added. Knowledge of this chapter will allow the reader to study recent literature on 7l. -codes. 4 Furthermore, some material has been added that appeared in my Springer Lec ture Notes 201, but was not included in earlier editions of this book, e. g. Generalized Reed-Solomon Codes and Generalized Reed-Muller Codes. In Chapter 2, a section on "Coding Gain" ( the engineer's justification for using error-correcting codes) was added. For the author, preparing this third edition was a most welcome return to mathematics after seven years of administration. For valuable discussions on the new material, I thank C.P.l.M.Baggen, I. M.Duursma, H.D.L.Hollmann, H. C. A. van Tilborg, and R. M. Wilson. A special word of thanks to R. A. Pellikaan for his assistance with Chapter 10. |
Sisältö
CHAPTER 1 | 1 |
CHAPTER 2 | 22 |
CHAPTER 3 | 33 |
CHAPTER 4 | 47 |
CHAPTER 5 | 64 |
CHAPTER 6 | 81 |
68 ReedSolomon Codes and Algebraic Geometry | 99 |
610 Binary Cyclic Codes of Length 2n n | 107 |
CHAPTER 10 | 148 |
107 Some Geometric Codes | 163 |
108 Improvement of the GilbertVarshamov Bound | 165 |
CHAPTER 11 | 167 |
CHAPTER 12 | 173 |
CHAPTER 13 | 181 |
133 An Analog of the Gilbert Bound for Some | 187 |
134 Construction of Convolutional Codes from Cyclic Block Codes | 189 |
CHAPTER 7 | 112 |
CHAPTER 8 | 128 |
CHAPTER 9 | 139 |
Hints and Solutions to Problems | 195 |
218 | |
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a₁ AG(m algebraic geometry b₁ BCH code binary code binary image block codes C₁ C₂ called Chapter code of length code over F codeword coding theory coefficients columns consider construction convolutional codes coordinates corresponding cosets curve cyclic code decoding defined Definition denote differential dimension divisor dual equation example F₂ field finite following theorem Golay code Goppa codes Hamming code Hence idempotent integer irreducible polynomials Lemma Let q linear code linear combination matrix G minimum distance multiplication nonzero nth root P₁ parameter parity check matrix perfect code permutation polynomial of degree positions primitive element PROOF QR code quaternary code rational points reader Reed-Muller code representation resp ring root of unity rows of G Section sequence subcode symbols Uniformly Packed Codes vector space weight enumerator words of weight x₁ yields zeros