Author | Lint, J. H. van. author |
---|---|
Title | Introduction to Coding Theory [electronic resource] / by J. H. van Lint |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1999 |
Edition | Third Revised and Expanded Edition |
Connect to | http://dx.doi.org/10.1007/978-3-642-58575-3 |
Descript | XIV, 234 p. online resource |
1 Mathematical Background -- 1.1. Algebra -- 1.2. Krawtchouk Polynomials -- 1.3. Combinatorial Theory -- 1.4. Probability Theory -- 2 Shannonโs Theorem -- 2.1. Introduction -- 2.2. Shannonโs 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. ReedโMuller 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. ReedโSolomon Codes -- 6.9. Quadratic Residue Codes -- 6.10. Binary Cyclic Codes of Length 2n(n odd) -- 6.11. Generalized ReedโMuller Codes -- 6.12. Comments -- 6.13. Problems -- 7 Perfect Codes and Uniformly Packed Codes -- 7.1. Lloydโs 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 RiemannโRoch Theorem -- 10.6. Codes from Algebraic Curves -- 10.7. Some Geometric Codes -- 10.8. Improvement of the GilbertโVarshamov 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. MandelbaumโBarrows 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