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ОбложкаCancellieri G. Polynomial theory of error correcting codes. - Cham: Springer, 2015. - xviii, 732 p.: ill., tab. - (Signals and communication technology). - Incl. bibl. ref. - Ind.: p.729-732. - ISBN 978-3-319-01726-6; ISSN 1860-4862
Шифр: (И/З.81-C19) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
                       Part I  Generator Matrix
1  Generator Matrix Approach to Linear Block Codes .............. 3
   1.1  Additive n × n Linear Transformation
   of a Binary Sequence ......................................... 3
   1.2  Generator Matrix G of a Linear Block Code ............... 5
   1.3  Polynomial Description of the Generator Matrix
        in a Linear Block Code ................................. 10
   1.4  Properties of a Linear Block Code Derived
        from the Structural Characteristics of g(x) ............ 15
   1.5  Systematic Encoder Circuit ............................. 19
   1.6  Code Concatenation: Effects on G Matrix ................ 21
   1.7  Code Puncturation: Effects on G Matrix ................. 28
   1.8  Cyclic Block Codes ..................................... 33
   1.9  Enumeration of all the Possible Cyclic Codes of
        Length N ............................................... 37
   1.10 Shortened Cyclic (SC) Codes ............................ 44
   1.11 Lengthened Cyclic (LC) Codes ........................... 47
   1.12 Subcode of an s.s. Time-Invariant Polynomial Code ...... 56
   1.13 Modified Lengthened Cyclic (MLC) Codes ................. 58
   1.14 State Diagrams ......................................... 61
   1.15 Direct Product Codes ................................... 68
   1.16 Generator Matrix of a Direct Product Code .............. 73
   1.17 Direct Product Codes as MLC Codes ...................... 75
   1.18 Interpretation of Particular Direct Product Codes
        by Means of GPC Codes .................................. 76
   1.19 Cyclic and Pseudo-Cyclic Codes in a Non-binary
        Alphabet ............................................... 79
   1.20 Q-ary State Diagrams ................................... 83
   1.21 Main Families of Non-binary Block Codes ................ 85
   1.22 Reed-Solomon Codes and Other MDS Non-binary Codes ...... 89
   1.23 Trellis for an s.s. Time-Invariant Block Code Obtained
        from Its Generator Matrix .............................. 95
   References .................................................. 98
2  Wide-Sense Time-Invariant Block Codes in Their
   Generator Matrix ........................................... 101
   2.1  Periodically Time-Varying Generator Matrix ............ 101
   2.2  Quasi-Cyclic Codes (QC) as a Widening in the Concept
        of Cyclic Codes ....................................... 105
   2.3  Quasi-Cyclic Codes with Distributed Control Symbols
        Described with Their G Matrix ......................... 112
   2.4  Representation of Known Block Codes as QC Codes
        with Distributed Control Symbols ...................... 117
   2.5  Relation Between Some Binary QC-Codes and Cyclic
        or Pseudo-Cyclic Codes in a Q-Ary Alphabet ............ 119
   2.6  Encoder Circuits Based on the G Matrix for a QC Code .  121
   2.7  Shortened Quasi-Cyclic (SQC) Codes .................... 125
   2.8  Lengthened Quasi-Cyclic (LQC) Codes ................... 130
   2.9  Subcode of a w.s. Time-Invariant Polynomial Code ...... 136
   2.10 Modified Lengthened Quasi-Cyclic (MLQC) Codes ......... 138
   2.11 Trellis for a w.s. Time-Invariant Block Code Obtained
        from Its Generator Matrix ............................. 143
   References ................................................. 147
3  Generator Matrix Approach to s.s. Time-Invariant
   Convolutional Codes ........................................ 149
   3.1  Traditional View of Non-systematic s.s.
        Time-Invariant Convolutional Codes .................... 149
   3.2  State Diagram and Minimum Distance .................... 154
   3.3  Systematic Convolutional Codes ........................ 162
   3.4  Low-Rate Convolutional Codes .......................... 166
   3.5  High-Rate Punctured Convolutional Codes ............... 172
   3.6  Recursive Systematic Convolutional (RSC) Codes ........ 176
   3.7  Equivalence Between MLC Codes and s.s. Time-Invariant
        Convolutional Codes ................................... 180
   3.8  Strict-Sense Time-Invariant High-Rate Convolutional
        (MLC) Codes ........................................... 183
   3.9  A First Bridge Between Cyclic Block Codes and s.s.
        Time-Invariant Convolutional Codes .................... 190
   3.10 Tail-Biting s.s. Time-Invariant Convolutional Codes ... 195
   3.11 Trellis of an s.s. Time-Invariant Convolutional
        Code Obtained from Its Generator Matrix ............... 199
   References ................................................. 203
4  Wide-Sense Time-Invariant Convolutional Codes
   in Their Generator Matrix .................................. 205
   4.1  Periodically Time-Varying Generator Matrix of
        a Convolutional Code .................................. 205
   4.2  Traditional Approach to w.s. Time-Invariant
        Convolutional Codes in Their G Matrix ................. 206
   4.3  Existence of the Inverse Linear Transformation ........ 215
   4.4  RSC Version of a w.s. Time-Invariant Convolutional
        Code .................................................. 218
   4.5  Equivalence Between MLQC Codes and a Certain Class
        of w.s. Time-Invariant Convolutional Codes ............ 222
   4.6  Practical Importance of Punctured Convolutional
        Codes ................................................. 225
   4.7  Tail-Biting w.s. Time-Invariant Convolutional Codes ... 226
   4.8  Unwrapping QC Block Codes and Reordered Versions of
        Convolutional Codes ................................... 231
   4.9  A First Bridge Between Quasi-cyclic Block Codes
        and w.s. Time-Invariant Convolutional Codes ........... 238
   4.10 Trellis of a w.s. Time-Invariant Convolutional Code
        Obtained from Its Generator Matrix .................... 240
   References ................................................. 242

                    Part II Parity Check Matrix
5  Parity Check Matrix Approach to Linear Block Codes ..........245
   5.1  Parity Check Matrix of a Linear Block Code ............ 245
   5.2  Parity Check Matrix and Hard-Decision Decoding ........ 249
   5.3  Relations Between the Parity Check Matrix
        and the Generator Matrix .............................. 255
   5.4  Polynomial Description of the Parity Check Matrix
        in a Linear Block Code ................................ 258
   5.5  Encoder Circuit Based on the Parity Check Polynomial
        and Its State Diagram ................................. 260
   5.6  Code Concatenation: Effects on H Matrix ............... 266
   5.7  Code Puncturation: Effects on H Matrix ................ 269
   5.8  Shortening Cyclic Codes: Effects on H Matrix .......... 272
   5.9  Lengthening Cyclic Codes: Effects on H Matrix ......... 274
   5.10 MLC Codes (s.s. Time-Invariant Convolutional Codes in
        G) and Their H Matrix ................................. 276
   5.11 Some Further Considerations About H Matrix in
        Polynomial Codes ...................................... 278
   5.12 Two Types of Rows in Polynomial Code H Matrix ......... 284
   5.13 H-Extended Cyclic (НЕС) Codes ......................... 286
   5.14 Discussion About Dual Polynomial Codes ................ 292
   5.15 Modified H-Extended Cyclic (MHEC) Codes ............... 297
   5.16 Direct Product Codes: Structure of Their H Matrix ..... 301
   5.17 Composite Codes Based on GPC Codes .................... 304
   5.18 H Matrix for Non-binary Block Codes ................... 312
   5.19 Trellis of an s.s. Time-Invariant Block Code
        Obtained from Its Parity Check Matrix ................. 315
   References ................................................. 319
6  Wide-Sense Time-Invariant Block Codes in Their Parity
   Check Matrix ........................................... 321
   6.1  Periodically Time-Varying Parity Check Matrix ......... 321
   6.2  Parity Check Matrix of a Quasi-Cyclic Code ............ 324
   6.3  Quasi-Cyclic Codes with Distributed Control Symbols
        Described with Their H Matrix ......................... 329
   6.4  Encoder Circuit Based on the H Matrix in a QC Code .... 332
   6.5  Shortened Quasi-Cyclic (SQC) Codes and Their H
        Matrix ................................................ 335
   6.6  Lengthened Quasi-Cyclic (LQC) Codes and Their H
        Matrix ................................................ 338
   6.7  Punctured QC Codes .................................... 341
   6.8  H-Extended Quasi-Cyclic (HEQC) Codes .................. 343
   6.9  Modified Lengthened Quasi-Cyclic (MLQC) Codes
        and Their H Matrix .................................... 346
   6.10 Modified H-Extended Quasi-Cyclic (MHEQC) Codes
        and Their H Matrix .................................... 347
   6.11 Trellis of a w.s. Time-Invariant Block Code Obtained
        from Its Parity Check Matrix .......................... 352
   References ................................................. 353
7  Strict-Sense Time-Invariant Convolutional Codes
   in Their Parity Check Matrix ............................... 355
   7.1  Syndrome Former Sub-matrix ............................ 355
   7.2  Construction of the Syndrome Former Sub-matrix
        for Low-Rate Convolutional Codes ...................... 360
   7.3  Extension of the Procedure for Obtaining the Syndrome
   Former Sub-matrix to High-Rate Convolutional Codes ......... 366
   7.4  Different Types of Not Well Designed Convolutional
        Codes ................................................. 372
   7.5  Interpretation of Direct Product Codes as Particular
        Not Well Designed Convolutional Codes ................. 381
   7.6  Various Situations for Well Designed and Not Well
        Designed Convolutional Codes Regarding Their Parity
        Check Matrix .......................................... 385
   7.7  Systematic Encoder Circuit Based on a Unique
        Non-periodic Parity Check Polynomial .................. 393
   7.8  Another Type of Systematic Encoder Circuit for s.s.
        Time-Invariant Convolutional Codes in Their H Matrix .. 398
   7.9  Tail-Biting Convolutional Codes s.s. Time-Invariant
        in H .................................................. 403
   7.10 Decoding Computational Complexity in the Trellis
        for s.s. Time-Invariant Convolutional Codes
        in Their H Matrix ..................................... 408
   7.11 A Second Bridge Between Cyclic Block Codes
        and s.s. Time-Invariant Convolutional Codes ........... 412
   References ................................................. 417
8  Wide-Sense Time-Invariant Convolutional Codes in Their
   Parity Check Matrix ........................................ 419
   8.1  Traditional Obtainment of a Symbolic Parity Check
        Matrix in a w.s. Time-Invariant Convolutional Code .... 419
   8.2  Null ex-OR Sum of Clusters of Syndromes for
        Obtaining G and Column Construction of H .............. 427
   8.3  Encoder Circuits Based on Different Parity Check
        Polynomials ........................................... 430
   8.4  Punctured Convolutional Codes and Their H Matrix ...... 432
   8.5  Tail-Biting w.s. Time-Invariant Convolutional
        Codes and Their H Matrix .............................. 434
   8.6  Unwrapping QC Codes Described by Their H Matrix ....... 436
   8.7  Reordered Versions of a w.s. Time-Invariant
        Convolutional Code .................................... 438
   8.8  Introductory Treatment of Array Codes ................. 443
   8.9  Generator Matrix of Improper Array Codes .............. 448
   8.10 A Second Bridge Between Quasi-Cyclic Codes
        and w.s. Time-Invariant Convolutional Codes ........... 454
   8.11 Unwrapping the Tail-Biting Convolutional Form
        of Improper Array Codes ............................... 462
   8.12 Further Considerations on Direct Product Codes
        Related to QC Array Codes ............................. 465
   References ................................................. 468

                      Part III   Modern Coding
9  Turbo Codes ................................................ 473
   9.1  The Basic Idea of Turbo Codes ......................... 473
   9.2  Some Particular Aspects of RSC Codes .................. 479
   9.3  Statistical Prediction of Turbo Code Performance ...... 482
   9.4  G Matrix of a Turbo Code and Correct
        Frame Termination ..................................... 483
   9.5  Outline of the Decoding Algorithm ..................... 488
   9.6  Turbo Codes in Serial Concatenation ................... 491
   9.7  Turbo-Product Codes ................................... 493
   9.8  Parity Check Matrix of Turbo Codes .................... 498
   References ................................................. 501
10 Low Density Parity Check Codes ............................. 503
   10.1 Tanner Graph and Message Passing Decoding
        Algorithms Constructed on It .......................... 503
   10.2 Short Cycles and the Row-Column Constraint ............ 506
   10.3 Main Families of LDPC Codes ........................... 512
   10.4 Masking and Row or Column Splitting ................... 517
   10.5 LDPC Codes Obtained from Superimposition .............. 521
   10.6 Procedures for Obtaining LDPC Codes from Known
        Non-LDPC Codes ........................................ 523
   10.7 Computer Based Design of Irregular LDPC Codes ......... 528
   10.8 Outline of the Sum-Product Algorithm, Its
        Computational Complexity and Points of Weakness ....... 530
   10.9 Statistical Analysis of the Asymptotic Behaviour
        of Regular and Irregular LDPC Codes ................... 533
   10.10 A First Approach to LDPC Convolutional Codes ......... 536
   References ................................................. 541
11 Binomial Product Generator LDPC Block Codes ................ 545
   11.1 Cyclic Version of a Composite Code Based on GPC
        Codes ................................................. 545
   11.2 Generator Polynomial for a Parallelepiped
        Concatenation of GPC Codes ............................ 549
   11.3 Evaluation of the Minimum Distance in BPG Block
        Codes ................................................. 553
   11.4 Effects of Combined Equalities and of Independent
        Equalities ............................................ 558
   11.5 Main Results on DC-GPC Codes and Direct Products
        of DC-GPC Codes ....................................... 562
   11.6 Improper Array LDPC Codes ............................. 567
   11.7 Generalized Array LDPC Codes .......................... 569
   11.8 Some Variants of Improper Array Codes ................. 572
   11.9 Single-Layer Parity Check Matrix QC Codes ............. 576
   References ................................................. 580
12 LDPC Convolutional Codes ................................... 581
   12.1 General Considerations ................................ 581
   12.2 Modified H-Extended Codes from BPG Block Codes ........ 585
   12.3 Convolutional LDPC Codes Obtained by Reordering
        and Unwrapping BPG Block Codes ........................ 590
   12.4 The Best Performance of LDPC Convolutional
        Codes up to Now ....................................... 595
   12.5 1/2-Rate LDPC Regular Convolutional Codes Directly
        Designed .............................................. 598
   12.6 High-Rate LDPC Regular Convolutional Codes Directly
        Designed .............................................. 603
   12.7 High-Rate Irregular LDPC Convolutional Codes
        Directly Designed ..................................... 609
   12.8 Low-Rate LDPC Convolutional Codes Directly Designed ... 612
   12.9 LDPC Convolutional Codes with Period Length and
        Number of Information Symbols Per Period not
        Co-prime .............................................. 617
   12.10 Final Comments ....................................... 619
   References ................................................. 621
Appendix A: Matrix Algebra in a Binary Finite Field ........... 623
Appendix B: Polynomial Representation of Binary Sequences ..... 643
Appendix C: Electronic Circuits for Multiplication or
            Division in Polynomial Representation of
            Binary Sequences .................................. 679
Appendix D: Survey on the Main Performance of Error
            Correcting Codes .................................. 691
Index ......................................................... 729

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