LEADER 06680nam 22006855 450 001 9910789214503321 005 20210913162330.0 010 $a3-540-64133-5 010 $a3-642-58575-2 024 7 $a10.1007/978-3-642-58575-3 035 $a(CKB)3400000000104512 035 $a(SSID)ssj0001297190 035 $a(PQKBManifestationID)11724801 035 $a(PQKBTitleCode)TC0001297190 035 $a(PQKBWorkID)11361993 035 $a(PQKB)11465315 035 $a(DE-He213)978-3-642-58575-3 035 $a(MiAaPQ)EBC3092660 035 $a(PPN)187451184 035 $a(EXLCZ)993400000000104512 100 $a20121227d1999 u| 0 101 0 $aeng 135 $aurnn#008mamaa 181 $ctxt 182 $cc 183 $acr 200 10$aIntroduction to Coding Theory$b[electronic resource] /$fby J.H. van Lint 205 $a3rd ed. 1999. 210 1$aBerlin, Heidelberg :$cSpringer Berlin Heidelberg :$cImprint: Springer,$d1999. 215 $a1 online resource (XIV, 234 p.) 225 1 $aGraduate Texts in Mathematics,$x0072-5285 ;$v86 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a3-540-78082-3 311 $a3-642-63653-5 320 $aIncludes bibliographical references and index. 327 $a1 Mathematical Background -- 1.1. Algebra -- 1.2. Krawtchouk Polynomials -- 1.3. Combinatorial Theory -- 1.4. Probability Theory -- 2 Shannon?s Theorem -- 2.1. Introduction -- 2.2. Shannon?s Theorem -- 2.3. On Coding Gain -- 2.4. Comments -- 2.5. Problems -- 3 Linear Codes -- 3.1. Block Codes -- 3.2. Linear Codes -- 3.3. Hamming Codes -- 3.4. Majority Logic Decoding -- 3.5. Weight Enumerators -- 3.6. The Lee Metric -- 3.7. Comments -- 3.8. Problems -- 4 Some Good Codes -- 4.1. Hadamard Codes and Generalizations -- 4.2. The Binary Golay Code -- 4.3. The Ternary Golay Code -- 4.4. Constructing Codes from Other Codes -- 4.5. Reed?Muller Codes -- 4.6. Kerdock Codes -- 4.7. Comments -- 4.8. Problems -- 5 Bounds on Codes -- 5.1. Introduction: The Gilbert Bound -- 5.2. Upper Bounds -- 5.3. The Linear Programming Bound -- 5.4. Comments -- 5.5. Problems -- 6 Cyclic Codes -- 6.1. Definitions -- 6.2. Generator Matrix and Check Polynomial -- 6.3. Zeros of a Cyclic Code -- 6.4. The Idempotent of a Cyclic Code -- 6.5. Other Representations of Cyclic Codes -- 6.6. BCH Codes -- 6.7. Decoding BCH Codes -- 6.8. Reed?Solomon Codes -- 6.9. Quadratic Residue Codes -- 6.10. Binary Cyclic Codes of Length 2n(n odd) -- 6.11. Generalized Reed?Muller Codes -- 6.12. Comments -- 6.13. Problems -- 7 Perfect Codes and Uniformly Packed Codes -- 7.1. Lloyd?s Theorem -- 7.2. The Characteristic Polynomial of a Code -- 7.3. Uniformly Packed Codes -- 7.4. Examples of Uniformly Packed Codes -- 7.5. Nonexistence Theorems -- 7.6. Comments -- 7.7. Problems -- 8 Codes over ?4 -- 8.1. Quaternary Codes -- 8.2. Binary Codes Derived from Codes over ?4 -- 8.3. Galois Rings over ?4 -- 8.4. Cyclic Codes over ?4 -- 8.5. Problems -- 9 Goppa Codes -- 9.1. Motivation -- 9.2. Goppa Codes -- 9.3. The Minimum Distance of Goppa Codes -- 9.4. Asymptotic Behaviour of Goppa Codes -- 9.5. Decoding Goppa Codes -- 9.6. Generalized BCH Codes -- 9.7. Comments -- 9.8. Problems -- 10 Algebraic Geometry Codes -- 10.1. Introduction -- 10.2. Algebraic Curves -- 10.3. Divisors -- 10.4. Differentials on a Curve -- 10.5. The Riemann?Roch Theorem -- 10.6. Codes from Algebraic Curves -- 10.7. Some Geometric Codes -- 10.8. Improvement of the Gilbert?Varshamov Bound -- 10.9. Comments -- 10.10.Problems -- 11 Asymptotically Good Algebraic Codes -- 11.1. A Simple Nonconstructive Example -- 11.2. Justesen Codes -- 11.3. Comments -- 11.4. Problems -- 12 Arithmetic Codes -- 12.1. AN Codes -- 12.2. The Arithmetic and Modular Weight -- 12.3. Mandelbaum?Barrows Codes -- 12.4. Comments -- 12.5. Problems -- 13 Convolutional Codes -- 13.1. Introduction -- 13.2. Decoding of Convolutional Codes -- 13.3. An Analog of the Gilbert Bound for Some Convolutional Codes -- 13.4. Construction of Convolutional Codes from Cyclic Block Codes -- 13.5. Automorphisms of Convolutional Codes -- 13.6. Comments -- 13.7. Problems -- Hints and Solutions to Problems -- References. 330 $aIt is gratifying that this textbook is still sufficiently popular to warrant a third edition. I have used the opportunity to improve and enlarge the book. When the second edition was prepared, only two pages on algebraic geometry codes were added. These have now been removed and replaced by a relatively long chapter on this subject. Although it is still only an introduction, the chapter requires more mathematical background of the reader than the remainder of this book. One of the very interesting recent developments concerns binary codes defined by using codes over the alphabet 7l.4? There is so much interest in this area that a chapter on the essentials was added. Knowledge of this chapter will allow the reader to study recent literature on 7l. -codes. 4 Furthermore, some material has been added that appeared in my Springer Lec­ ture Notes 201, but was not included in earlier editions of this book, e. g. Generalized Reed-Solomon Codes and Generalized Reed-Muller Codes. In Chapter 2, a section on "Coding Gain" ( the engineer's justification for using error-correcting codes) was added. For the author, preparing this third edition was a most welcome return to mathematics after seven years of administration. For valuable discussions on the new material, I thank C.P.l.M.Baggen, I. M.Duursma, H.D.L.Hollmann, H. C. A. van Tilborg, and R. M. Wilson. A special word of thanks to R. A. Pellikaan for his assistance with Chapter 10. 410 0$aGraduate Texts in Mathematics,$x0072-5285 ;$v86 606 $aCombinatorics 606 $aAlgebraic geometry 606 $aNumber theory 606 $aCombinatorics$3https://scigraph.springernature.com/ontologies/product-market-codes/M29010 606 $aAlgebraic Geometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M11019 606 $aNumber Theory$3https://scigraph.springernature.com/ontologies/product-market-codes/M25001 615 0$aCombinatorics. 615 0$aAlgebraic geometry. 615 0$aNumber theory. 615 14$aCombinatorics. 615 24$aAlgebraic Geometry. 615 24$aNumber Theory. 676 $a003/.54 686 $a94A24$2msc 700 $aLint$b J.H. van$4aut$4http://id.loc.gov/vocabulary/relators/aut$055409 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910789214503321 996 $aIntroduction to coding theory$9374924 997 $aUNINA