Alain Glavieux, was an Associate Professor at the Ecole Nationale Supérieure des Télécommunications de Bretagne (ENST Bretagne), where he lectured in signal processing and communication theory and set up a digital communications research group. He was appointed Professor in 1989, and became Head of the Signals and Communications Department in 1994. In 1998, he took charge of ENST Bretagne's Corporate Relations with Industry and, in 2002, became Head of the CNRS laboratory Algorithmic and Hardware Processing of Information, Communications and Knowledge. In 2003, he was appointed assistant director of ENST Bretagne.Together with Professor Claude Berrou, Alain Glavieux developed a new family of error correction codes called turbo codes. He also pioneered the principle of turbo equalization.He received an IEEE Golden Jubilee Award for Technological Innovation in 1998, the IEEE Richard W. Hamming Medal in 2003, and the French Academy of Sciences Grand Prix France Telecom in 2003.He passed away after a long illness on September 25th, 2004 at the age of 55.

"The book offers a very good overview of channel coding topics, selected and proposed by a group of experienced researchers including inventors of turbocodes from ENST Bretagne . . . The presented material is well illustrated with examples, referring to both encoding and decoding schemes . . . The most valuable part of the book consists of chapters from 3 to 7 and is connected with convolutional codes, trellis coded modulations and especially turbocodes." (Einzelbeleg, 2010) "This book provides a comprehensive overview of the subject of channel coding . . . as such, anyone involved in the area of channel coding and error correcting coding will find this book to be of invaluable assistance." (Mathematical Reviews, 2010)

• Homage to Alain Glavieux xvChapter 1. Information Theory 1Gérard BATTAIL1.1. Introduction: the Shannon paradigm 11.2. Principal coding functions 51.2.1. Source coding 51.2.2. Channel coding 61.2.3. Cryptography 71.2.4. Standardization of the Shannon diagram blocks 81.2.5. Fundamental theorems 91.3. Quantitative measurement of information 91.3.1. Principle 91.3.2. Measurement of self-information 101.3.3. Entropy of a source 111.3.4. Mutual information measure 121.3.5. Channel capacity 141.3.6. Comments on the measurement of information 151.4. Source coding 151.4.1. Introduction 151.4.2. Decodability, Kraft-McMillan inequality 161.4.3. Demonstration of the fundamental theorem 171.4.4. Outline of optimal algorithms of source coding 181.5. Channel coding 191.5.1. Introduction and statement of the fundamental theorem 191.5.2. General comments 201.5.3. Need for redundancy 201.5.4. Example of the binary symmetric channel 211.5.5. A geometrical interpretation 251.5.6. Fundamental theorem: Gallager’s proof 261.6. Channels with continuous noise 321.6.1. Introduction 321.6.2. A reference model in physical reality: the channel with Gaussian additive noise 321.6.3. Communication via a channel with additive white Gaussian noise 351.6.4. Channel with fadings 371.7. Information theory and channel coding 381.8. Bibliography 40Chapter 2. Block Codes 41Alain POLI2.1. Unstructured codes 412.1.1. The fundamental question of message redundancy 412.1.2. Unstructured codes 422.2. Linear codes 442.2.1. Introduction 442.2.2. Properties of linear codes 442.2.3. Dual code 462.2.4. Some linear codes 502.2.5. Decoding of linear codes 512.3. Finite fields 532.3.1. Basic concepts 532.3.2. Polynomial modulo calculations: quotient ring 532.3.3. Irreducible polynomial modulo calculations: finite field 542.3.4. Order and the opposite of an element of F2[X]/(p(X)) 542.3.5. Minimum polynomials 592.3.6. The field of nth roots of unity 602.3.7. Projective geometry in a finite field 612.4. Cyclic codes 622.4.1. Introduction 622.4.2. Base, coding, dual code and code annihilator 632.4.3. Certain cyclic codes 682.4.4. Existence and construction of cyclic codes 742.4.5. Applications of cyclic codes 822.5. Electronic circuits 822.5.1. Basic gates for error correcting codes 822.5.2. Shift registers 832.5.3. Circuits for the correct codes 832.5.4. Polynomial representation and representation to the power of a primitive representation for a field 872.6. Decoding of cyclic codes 882.6.1. Meggitt decoding (trapping of bursts) 882.6.2. Decoding by the DFT 892.6.3. FG-decoding 942.6.4. Berlekamp-Massey decoding 992.6.5. Majority decoding 1052.6.6. Hard decoding, soft decoding and chase decoding 1102.7. 2D codes 1112.7.1. Introduction 1112.7.2. Product codes 1122.7.3. Minimum distance of 2D codes 1122.7.4. Practical examples of the use of 2D codes 1122.7.5. Coding 1122.7.6. Decoding 1132.8. Exercises on block codes 1132.8.1. Unstructured codes 1132.8.2. Linear codes 1142.8.3. Finite bodies 1172.8.4. Cyclic codes 1192.8.5. Exercises on circuits 123Chapter 3. Convolutional Codes 129Alain GLAVIEUX and Sandrine VATON3.1. Introduction 1293.2. State transition diagram, trellis, tree 1353.3. Transfer function and distance spectrum 1373.4. Perforated convolutional codes 1403.5. Catastrophic codes 1423.6. The decoding of convolutional codes 1423.6.1. Viterbi algorithm 1433.6.2. MAP criterion or BCJR algorithm 1563.6.3. SubMAP algorithm 1693.7. Performance of convolutional codes 1723.7.1. Channel with binary input and continuous output 1733.7.2. Channel with binary input and output 1803.8. Distance spectrum of convolutional codes 1823.9. Recursive convolution codes 184Chapter 4. Coded Modulations 197Ezio BIGLIERI4.1. Hamming distance and Euclidean distance 1974.2. Trellis code 2004.3. Decoding 2014.4. Some examples of TCM 2014.5. Choice of a TCM diagram 2054.6. TCM representations 2074.7. TCM transparent to rotations 2094.7.1. Partitions transparent to rotations 2114.7.2. Transparent trellis with rotations 2124.7.3. Transparent encoder 2134.7.4. General considerations 2154.8. TCM error probability 2154.8.1. Upper limit of the probability of an error event 2154.8.2. Examples 2264.8.3. Calculation of áfree 2284.9. Power spectral density 2324.10. Multi-level coding 2344.10.1. Block coded modulation 2354.10.2. Decoding of multilevel codes by stages 2374.11. Probability of error for the BCM 2384.11.1. Additive Gaussian channel 2394.11.2. Calculation of the transfer function 2404.12. Coded modulations for channels with fading 2414.12.1. Modeling of channels with fading 2414.12.2. Rayleigh fading channel: Euclidean distance and Hamming distance 2474.13. Bit interleaved coded modulation (BICM) 2514.14. Bibliography 253Chapter 5. Turbocodes 255Claude BERROU, Catherine DOUILLARD, Michel JÉZÉQUEL and Annie PICART5.1. History of turbocodes 2555.1.1. Concatenation 2565.1.2. Negative feedback in the decoder 2565.1.3. Recursive systematic codes 2585.1.4. Extrinsic information 2585.1.5. Parallel concatenation 2595.1.6. Irregular interleaving 2605.2. A simple and convincing illustration of the turbo effect 2605.3. Turbocodes 2655.3.1. Coding 2655.3.2. The termination of constituent codes 2725.3.3. Decoding 2755.3.4. SISO decoding and extrinsic information 2805.4. The permutation function 2875.4.1. The regular permutation 2885.4.2. Statistical approach 2905.4.3. Real permutations 2915.5. m-binary turbocodes 2975.5.1. m-binary RSC encoders 2985.5.2. m-binary turbocodes 3005.5.3. Double-binary turbocodes with 8 states 3025.5.4. Double-binary turbocodes with 16 states 3035.6. Bibliography 304Chapter 6. Block Turbocodes 307Ramesh PYNDIAH and Patrick ADDE6.1. Introduction 3076.2. Concatenation of block codes 3086.2.1. Parallel concatenation of block codes 3096.2.2. Serial concatenation of block codes 3136.2.3. Properties of product codes and theoretical performances 3186.3. Soft decoding of block codes 3236.3.1. Soft decoding of block codes 3246.3.2. Soft decoding of block codes (Chase algorithm) 3266.3.3. Decoding of block codes by the Viterbi algorithm 3346.3.4. Decoding of block codes by the Hartmann and Rudolph algorithm 3386.4. Iterative decoding of product codes 3406.4.1. SISO decoding of a block code 3416.4.2. Implementation of the weighting algorithm 3456.4.3. Iterative decoding of product codes 3476.4.4. Comparison of the performances of BTC 3496.5. Conclusion 3676.6. Bibliography 367Chapter 7. Block Turbocodes in a Practical Setting 373Patrick ADDE and Ramesh PYNDIAH7.1. Introduction 3737.2. Implementation of BTC: structure and complexity 3737.2.1. Influence of integration constraints 3737.2.2. General architecture and organization of the circuit 3767.2.3. Memorizing of data and results 3807.2.4. Elementary decoder 3847.2.5. High flow structure 3927.3. Flexibility of turbo block codes 3977.4. Hybrid turbocodes 4047.4.1. Construction of the code 4047.4.2. Binary error rates (BER) function of the signal-to-noise ratio in a Gaussian channel 4067.4.3. Variation of the size of the blocks 4087.4.4. Variation of the total rate 4097.5. Multidimensional turbocodes 4097.6. Bibliography 412List of Authors 415Index 417

Alain Glavieux, was an Associate Professor at the Ecole Nationale Supérieure des Télécommunications de Bretagne (ENST Bretagne), where he lectured in signal processing and communication theory and set up a digital communications research group. He was appointed Professor in 1989, and became Head of the Signals and Communications Department in 1994. In 1998, he took charge of ENST Bretagne's Corporate Relations with Industry and, in 2002, became Head of the CNRS laboratory Algorithmic and Hardware Processing of Information, Communications and Knowledge. In 2003, he was appointed assistant director of ENST Bretagne.Together with Professor Claude Berrou, Alain Glavieux developed a new family of error correction codes called turbo codes. He also pioneered the principle of turbo equalization.He received an IEEE Golden Jubilee Award for Technological Innovation in 1998, the IEEE Richard W. Hamming Medal in 2003, and the French Academy of Sciences Grand Prix France Telecom in 2003.He passed away after a long illness on September 25th, 2004 at the age of 55.

"The book offers a very good overview of channel coding topics, selected and proposed by a group of experienced researchers including inventors of turbocodes from ENST Bretagne . . . The presented material is well illustrated with examples, referring to both encoding and decoding schemes . . . The most valuable part of the book consists of chapters from 3 to 7 and is connected with convolutional codes, trellis coded modulations and especially turbocodes." (Einzelbeleg, 2010) "This book provides a comprehensive overview of the subject of channel coding . . . as such, anyone involved in the area of channel coding and error correcting coding will find this book to be of invaluable assistance." (Mathematical Reviews, 2010)

• Homage to Alain Glavieux xvChapter 1. Information Theory 1Gérard BATTAIL1.1. Introduction: the Shannon paradigm 11.2. Principal coding functions 51.2.1. Source coding 51.2.2. Channel coding 61.2.3. Cryptography 71.2.4. Standardization of the Shannon diagram blocks 81.2.5. Fundamental theorems 91.3. Quantitative measurement of information 91.3.1. Principle 91.3.2. Measurement of self-information 101.3.3. Entropy of a source 111.3.4. Mutual information measure 121.3.5. Channel capacity 141.3.6. Comments on the measurement of information 151.4. Source coding 151.4.1. Introduction 151.4.2. Decodability, Kraft-McMillan inequality 161.4.3. Demonstration of the fundamental theorem 171.4.4. Outline of optimal algorithms of source coding 181.5. Channel coding 191.5.1. Introduction and statement of the fundamental theorem 191.5.2. General comments 201.5.3. Need for redundancy 201.5.4. Example of the binary symmetric channel 211.5.5. A geometrical interpretation 251.5.6. Fundamental theorem: Gallager’s proof 261.6. Channels with continuous noise 321.6.1. Introduction 321.6.2. A reference model in physical reality: the channel with Gaussian additive noise 321.6.3. Communication via a channel with additive white Gaussian noise 351.6.4. Channel with fadings 371.7. Information theory and channel coding 381.8. Bibliography 40Chapter 2. Block Codes 41Alain POLI2.1. Unstructured codes 412.1.1. The fundamental question of message redundancy 412.1.2. Unstructured codes 422.2. Linear codes 442.2.1. Introduction 442.2.2. Properties of linear codes 442.2.3. Dual code 462.2.4. Some linear codes 502.2.5. Decoding of linear codes 512.3. Finite fields 532.3.1. Basic concepts 532.3.2. Polynomial modulo calculations: quotient ring 532.3.3. Irreducible polynomial modulo calculations: finite field 542.3.4. Order and the opposite of an element of F2[X]/(p(X)) 542.3.5. Minimum polynomials 592.3.6. The field of nth roots of unity 602.3.7. Projective geometry in a finite field 612.4. Cyclic codes 622.4.1. Introduction 622.4.2. Base, coding, dual code and code annihilator 632.4.3. Certain cyclic codes 682.4.4. Existence and construction of cyclic codes 742.4.5. Applications of cyclic codes 822.5. Electronic circuits 822.5.1. Basic gates for error correcting codes 822.5.2. Shift registers 832.5.3. Circuits for the correct codes 832.5.4. Polynomial representation and representation to the power of a primitive representation for a field 872.6. Decoding of cyclic codes 882.6.1. Meggitt decoding (trapping of bursts) 882.6.2. Decoding by the DFT 892.6.3. FG-decoding 942.6.4. Berlekamp-Massey decoding 992.6.5. Majority decoding 1052.6.6. Hard decoding, soft decoding and chase decoding 1102.7. 2D codes 1112.7.1. Introduction 1112.7.2. Product codes 1122.7.3. Minimum distance of 2D codes 1122.7.4. Practical examples of the use of 2D codes 1122.7.5. Coding 1122.7.6. Decoding 1132.8. Exercises on block codes 1132.8.1. Unstructured codes 1132.8.2. Linear codes 1142.8.3. Finite bodies 1172.8.4. Cyclic codes 1192.8.5. Exercises on circuits 123Chapter 3. Convolutional Codes 129Alain GLAVIEUX and Sandrine VATON3.1. Introduction 1293.2. State transition diagram, trellis, tree 1353.3. Transfer function and distance spectrum 1373.4. Perforated convolutional codes 1403.5. Catastrophic codes 1423.6. The decoding of convolutional codes 1423.6.1. Viterbi algorithm 1433.6.2. MAP criterion or BCJR algorithm 1563.6.3. SubMAP algorithm 1693.7. Performance of convolutional codes 1723.7.1. Channel with binary input and continuous output 1733.7.2. Channel with binary input and output 1803.8. Distance spectrum of convolutional codes 1823.9. Recursive convolution codes 184Chapter 4. Coded Modulations 197Ezio BIGLIERI4.1. Hamming distance and Euclidean distance 1974.2. Trellis code 2004.3. Decoding 2014.4. Some examples of TCM 2014.5. Choice of a TCM diagram 2054.6. TCM representations 2074.7. TCM transparent to rotations 2094.7.1. Partitions transparent to rotations 2114.7.2. Transparent trellis with rotations 2124.7.3. Transparent encoder 2134.7.4. General considerations 2154.8. TCM error probability 2154.8.1. Upper limit of the probability of an error event 2154.8.2. Examples 2264.8.3. Calculation of áfree 2284.9. Power spectral density 2324.10. Multi-level coding 2344.10.1. Block coded modulation 2354.10.2. Decoding of multilevel codes by stages 2374.11. Probability of error for the BCM 2384.11.1. Additive Gaussian channel 2394.11.2. Calculation of the transfer function 2404.12. Coded modulations for channels with fading 2414.12.1. Modeling of channels with fading 2414.12.2. Rayleigh fading channel: Euclidean distance and Hamming distance 2474.13. Bit interleaved coded modulation (BICM) 2514.14. Bibliography 253Chapter 5. Turbocodes 255Claude BERROU, Catherine DOUILLARD, Michel JÉZÉQUEL and Annie PICART5.1. History of turbocodes 2555.1.1. Concatenation 2565.1.2. Negative feedback in the decoder 2565.1.3. Recursive systematic codes 2585.1.4. Extrinsic information 2585.1.5. Parallel concatenation 2595.1.6. Irregular interleaving 2605.2. A simple and convincing illustration of the turbo effect 2605.3. Turbocodes 2655.3.1. Coding 2655.3.2. The termination of constituent codes 2725.3.3. Decoding 2755.3.4. SISO decoding and extrinsic information 2805.4. The permutation function 2875.4.1. The regular permutation 2885.4.2. Statistical approach 2905.4.3. Real permutations 2915.5. m-binary turbocodes 2975.5.1. m-binary RSC encoders 2985.5.2. m-binary turbocodes 3005.5.3. Double-binary turbocodes with 8 states 3025.5.4. Double-binary turbocodes with 16 states 3035.6. Bibliography 304Chapter 6. Block Turbocodes 307Ramesh PYNDIAH and Patrick ADDE6.1. Introduction 3076.2. Concatenation of block codes 3086.2.1. Parallel concatenation of block codes 3096.2.2. Serial concatenation of block codes 3136.2.3. Properties of product codes and theoretical performances 3186.3. Soft decoding of block codes 3236.3.1. Soft decoding of block codes 3246.3.2. Soft decoding of block codes (Chase algorithm) 3266.3.3. Decoding of block codes by the Viterbi algorithm 3346.3.4. Decoding of block codes by the Hartmann and Rudolph algorithm 3386.4. Iterative decoding of product codes 3406.4.1. SISO decoding of a block code 3416.4.2. Implementation of the weighting algorithm 3456.4.3. Iterative decoding of product codes 3476.4.4. Comparison of the performances of BTC 3496.5. Conclusion 3676.6. Bibliography 367Chapter 7. Block Turbocodes in a Practical Setting 373Patrick ADDE and Ramesh PYNDIAH7.1. Introduction 3737.2. Implementation of BTC: structure and complexity 3737.2.1. Influence of integration constraints 3737.2.2. General architecture and organization of the circuit 3767.2.3. Memorizing of data and results 3807.2.4. Elementary decoder 3847.2.5. High flow structure 3927.3. Flexibility of turbo block codes 3977.4. Hybrid turbocodes 4047.4.1. Construction of the code 4047.4.2. Binary error rates (BER) function of the signal-to-noise ratio in a Gaussian channel 4067.4.3. Variation of the size of the blocks 4087.4.4. Variation of the total rate 4097.5. Multidimensional turbocodes 4097.6. Bibliography 412List of Authors 415Index 417