Table of contents for Error-correcting linear codes : classification by isometry and applications / Anton Betten ... [et al.].


Bibliographic record and links to related information available from the Library of Congress catalog
Note: Electronic data is machine generated. May be incomplete or contain other coding.


Counter
1       Linear Codes
1.1     Introduction ............... .. . ............... ..... ...........  3
1.2     Linear Codes, Encoding  and  Decoding......................  11
1.3     Check Matrices and the Dual Code......................  20
1.4     Classification  by  Isom etry  ........................ ...........  28
1.5     Semilinear Isometry Classes of Linear Codes ............  41
1.6     The Weight Enumerator ...............  ...............  51
1.7     Systematic Encoding, Information Sets ...................  65
1.8     A  Minimum  Distance  Algorithm ............................  70
2       Bounds and Modifications
2.1     Combinatorial Bounds for the Parameters................  82
2.2     New Codes from Old and the Minimum Distance .......     94
2.3     Further Modifications and Constructions ..................  102
2.4     Reed-Muller-Codes.......  ....... .. ............................  118
2.5     MDS-Codes ..................................................  128
3       Finite Fields
3.1     Finite  Fields -  An  Introduction  ........ ...... ... ...........  139
3.2     Existence and Uniqueness of Finite Fields .............  149
3.3     The Galois Group and Normal Bases....................  167
3.4     Enumeration under Group Actions, Lyndon Words.......   170
3.5     Construction of Irreducible Polynomials...................  182
3.6     Representations of Field  Elements ..........................  203
3.7     Projective  Geom etry  ............................. .............  205
4       Cyclic Codes
4.1     Cyclic Codes as Group Algebra Codes.....................  214
4.2     Polynomial Representation of Cyclic Codes ..............  220
4.3     BCH-Codes and Reed-Solomon-Codes.....................  237
4.4     Quadratic-Residue-Codes, Golay-Codes.................... 252
4.5    Idempotents and the Discrete Fourier Transform ......... 268
4.6    Alternant-Codes, Goppa-Codes ........................... 285
4.7    The  Structure Theorem  ......................................  292
4.8     Codes of p-Power Block  Length.............................  311
4.9     Bounds for the  Minimum  Distance...........................  319
4.10    Reed-Muller-Codes.................. ..............  327
4.11    Encoding  .................... ... .. ........... . ...... ....  334
4.12    Permutation Decoding................................... 338
4.13    Error-Correcting Pairs  .............. ............ 346
4.14    Majority  Logic  Decoding ................ ...................  350
5       Mathematics and Audio Compact Discs
5.1     Fourier Transform, Shannon's Sampling Theorem ........ 370
5.2     Correction  of Erasures........................ ...............  389
5.3     Burst Errors and Interleaving of Codes.................... 401
5.4     More Details on Compact Discs..........................  423
5.5     More  Details on  CD-ROM  ....................................  435
6       Enumeration of Isometry Classes
6.1     Enumeration of Linear Isometry Classes .................. 444
6.2     Indecomposable  Linear Codes..............................  463
6.3    Cycle Indices of Projective Linear Groups ................. 476
6.4     Numerical Data for Linear Isometry Classes .............. 499
6.5     Critical Codes  ................. ....  .  ...... .......... .  511
6.6     Random Generation of Linear Codes ....................... 527
6.7     Enumeration of Semilinear Isometry Classes .............. 532
6.8     Local Isom etries............................. ....... ..... .  549
6.9     Existence and Construction of Normal Bases............. 553
7      Solving Systems of Diophantine Linear Equations
7.1     Lattices ............... . ....................... .............. .  565
7.2     Diophantine Equations and  Lattices........................  568
7.3     Basic  Theory  of Lattices ...................... ..............  574
7.4    Gram-Schmidt Orthogonalization........................... 577
7.5     Bounds on  Lattice  Vectors..................... ..............  579
7.6     Lattice  Basis  Reduction  .............. . ... ...............  586
7.7     Lattice  Point  Enumeration.......... ...... ...............  598
7.8    Computing the Minimum Distance of Linear Codes...... 605
8       Linear Codes with a Prescribed Minimum Distance
8.1     Minihypers ....................................................  616
8.2     Group  Actions on  Lattices ............... . ..............  625
8.3      Prescribing a Group of Automorphisms .....................  637
8.4      Linear Codes of Prescribed  Type  .............................  640
8.5      Numerical Results ..........................................  644
9        The General Case
9.1      T he  Problem ............... ............  ...............   664
9.2      Computing with Permutation Groups .......................   669
9.3      A Permutation Representation ..............      ...........  676
9.4      The  Lexicographical O rder....................................  682
9.5      Orderly  Generation  of Codes ................................  688
9.6      The Algorithm Snakes and Ladders..........................  700
9.7      Base and Strong Generating Sets .........................   717
9.8      The Projective Linear Group ................................  727
9.9      The Projective Semilinear Group .............................  738
9.10     Numerical Data ............... .........      ............  741
A        Appendix: The Attached Compact Disc
A.1      System  Requirem  ents .........................................  755
A.2      The Installation ................ ............................  755
A.3      The Programs ................................................  756
A.4      The Dynamic Tables .........................................  757
A.5      The Precomputed Tables: Enumerative Results...........      758
A.6      The Precomputed Tables: Optimal Linear Codes.......         759
A.7      The Programs for Chapter 9 ...............................  763



Library of Congress subject headings for this publication: Combinatorial analysis, Error-correcting codes (Information theory)Coding theory