Distance-Preserving Mappings and Trellis Codes with Permutation Sequences by Theodorus Gerhardus Swart A thesis submitted to the Faculty of Engineering and the Built Environment in the fulfillment of the requirements for the degree of Doctor Ingeneriae in Electric and Electronic Engineering Science at the University of Johannesburg Promoter Prof. H.C. Ferreira August, 2006 Abstract Our research is focused on mapping binary sequences to permutation sequences. It is es- tablished that an upper bound on the sum of the Hamming distance for all mappings ex- ists, and this sum is used as a criterion to ascertain how good previously known mappings are. We further make use of permutation trellis codes to investigate the performance of certain permutation mappings in a power-line communications system, where background noise, narrow band noise and wide band noise are present. A new multilevel construction is presented next that maps binary sequences to per- mutation sequences, creating new mappings for which the sum of Hamming distances are greater than previous known mappings. It also proved that for certain lengths of sequences, the new construction can attain our new upper bound on the sum of Hamming distances. We further extend the multilevel construction by showing how it can be applied to other mappings, such as permutations with repeating symbols and mappings with non- binary inputs. We also show that a subset of the new construction yields permutation sequences that are able to correct insertion and deletion errors as well. Finally, we show that long binary sequences, formed by concatenating the columns of binary permutation matrices, are subsets of the Levenshtein insertion/deletion correcting codes. ii Opsomming Ons navorsing is gefokus op die afbeelding van binˆeresekwensies na permutasie sekwensies. Dit word aangetoon dat daar ’n bo-grens op die som van die Hamming afstande in so ’n afbeelding bestaan, en hierdie som word gebruik as maatstaf om te bepaal hoe effektief bestaande afbeeldings is. Verder maak ons gebruik van permutasie traliekodes om die gedrag van sekere permutasie afbeeldings te ondersoek in ’n kraglyn kommunikasie stelsel, waar agtergrond geraas, nouband geraas en wyeband geraas teenwoordig is. ’n Nuwe multivlak konstruksie word aangebied wat binˆeresekwensies na permutasie sekwensies afbeeld, wat nuwe afbeeldings skep waarvoor die som van Hamming afstande groter is as bestaande afbeeldings. Dit word ook bewys dat vir sekere lengtes van sek- wensies, die nuwe konstruksie die bo grens op die som van Hamming afstande kan haal. Ons brei die multivlak konstruksie verder uit deur te wys dat dit op ander afbeeldings ook toegepas kan word, soos permutasies met herhaalde simbole en afbeeldings met nie- binˆereinsette. Ons wys ook dat ’n deelversameling van die nuwe konstruksie permutasie sekwensies lewer wat invoegings en weglatings ook kan korrigeer. Laastens wys ons dat lang binˆeresekwensies, gevorm deur die kolomme van binˆere permutasie matrikse te kombineer, ’n deelversameling is van die Levenshtein invoeg- ings/weglatings kodes. iii Acknowledgements A sincere word of thanks to . • Prof. Ferreira for his help and guidance throughout the course of this study, • my fellow students in the Cybernetics Laboratory at the University of Johannesburg for discussions and help, • my family and friends for their interest, • the NRF for financial support, as well as the NRF/Department of Labour for finan- cial support through the Scarce Skills bursary, • Ken Stewart for proofreading the thesis, and • the examination committee, Prof. Abdel-Ghaffar, Prof. Penzhorn and Prof. Vinck for reading the thesis and for their comments. iv Table of Contents List of Figures . x List of Tables . xii List of Symbols . xiv 1 Introduction . 1-1 1.1 Introduction . 1-1 1.2 Problem statement . 1-2 1.3 Outline of thesis . 1-2 2 Overview of Permutation Codes . 2-1 2.1 Introduction . 2-1 2.2 Permutations in general . 2-1 2.2.1 Mathematical definition . 2-1 2.2.2 Slepian permutation codes . 2-3 2.2.3 Other related work . 2-4 2.3 Permutation arrays . 2-5 2.4 Distance-preserving mappings . 2-8 v Table of Contents 2.5 Distance-preserving permutation mappings . 2-9 2.6 Permutation codes and modulation . 2-11 2.7 Permutation source coding and quantizers . 2-14 2.8 Other applications of permutations . 2-14 3 Overview of Insertion/Deletion Correcting Codes . 3-1 3.1 Introduction . 3-1 3.2 Synchronization and insertions/deletions . 3-1 3.3 Block codes . 3-4 3.3.1 Synchronization recovery . 3-4 3.3.2 Error correction . 3-8 3.4 Convolutional codes . 3-13 3.5 Other applications . 3-14 4 Permutation Trellis Codes . 4-1 4.1 Introduction . 4-1 4.2 Permutation codes, M-ary FSK and PLC . 4-2 4.3 Permutation trellis codes and distance-preserving mappings . 4-4 4.4 Simulation setup . 4-7 4.5 Preliminary performance results . 4-8 4.6 Summary . 4-13 5 Distance Optimality of Permutation and Other Mappings . 5-1 5.1 Introduction . 5-1 5.2 Distance-preserving mappings . 5-1 5.3 Upper bound on the sum of distances in permutation mappings . 5-4 vi Table of Contents 5.4 General upper bound on the sum of distances in a mapping . 5-12 5.4.1 q-ary (n, k) codes . 5-15 5.4.2 M(n, M, δ) permutation codes . 5-18 5.5 Average distance increase in a mapping . 5-19 5.6 Summary . 5-22 6 Multilevel Construction of Permutation DPMs . 6-1 6.1 Introduction . 6-1 6.2 Previous constructions . 6-1 6.3 Binary representation of any permutation . 6-3 6.4 Generating SM from the multilevel construction . 6-6 6.5 DPMs from the multilevel construction . 6-11 6.6 Distance optimality of new DPMs . 6-26 6.7 Summary . 6-29 7 Comparison and Analysis of Different Mappings . 7-1 7.1 Introduction . 7-1 7.2 Mappings used in PLC simulations . 7-1 7.3 Mappings from the multilevel construction . 7-7 7.4 Permutation DPM constructions . 7-9 7.5 Graph representation of permutation mappings . 7-13 7.5.1 Graph representation of symbols and transpositions . 7-14 7.5.2 Graph representation of permutation DPMs . 7-15 7.5.3 Symbol distribution and symbol paths . 7-18 7.6 Summary . 7-24 vii Table of Contents 8 Multilevel Construction of Other Mappings . 8-1 8.1 Introduction . 8-1 8.2 Permutations with repeating symbols . 8-1 8.3 Non-binary sequences to permutation sequences . 8-4 8.4 Single insertion/deletion correcting codes . 8-6 8.5 Summary . 8-12 9 Binary Permutation Codes as Subsets of Levenshtein Codes . 9-1 9.1 Introduction . 9-1 9.2 Preliminaries . 9-1 9.3 Sequences with λ =1............................. 9-3 9.4 Other λ sequences as permutation matrices . 9-5 9.5 Sequences with any λ ............................. 9-7 9.6 Subsets of other codes . 9-9 9.7 Summary . 9-13 10 Conclusion . 10-1 10.1 Achievements . 10-1 10.2 Further research . 10-3 A Convolutional Codes used for Simulations . A-1 B Permutation Distance-Preserving Mappings . B-1 B.1 Permutation distance-conserving mappings . B-1 B.2 Permutation distance-increasing mappings . B-3 B.3 Permutation distance-reducing mappings . B-4 viii Table of Contents C Other Distance-Preserving Mappings . C-1 C.1 Other distance-conserving mappings . C-1 C.2 Other distance-increasing mappings . C-3 C.3 Other distance-reducing mappings . C-4 D Graph Representation of Permutation Mappings . D-1 References . Rf-1 ix List of Figures 3.1 Effect of deletion(s) and insertion(s) on data . 3-2 4.1 State diagrams for convolutional base code and permutation trellis code . 4-5 4.2 BER for background noise . 4-9 4.3 BER for impulse noise . 4-10 4.4 BER for background noise with permanent frequency disturbances . 4-10 4.5 Single permanent frequency disturbance in different positions . 4-14 4.6 Comparison of various number of permanent frequency disturbances . 4-15 6.1 Comparison of mapping using standard lexicography and mapping using the multilevel construction . 6-13 7.1 Comparison of optimum and non-optimum mappings, with background noise and permanent frequency disturbances . 7-4 7.2 Visualisation of distance distribution for different M(6, 6, 0) mappings . 7-7 7.3 Distance increase distribution for four different C3 components . 7-9 7.4 Distance optimality, η, for various mappings . 7-11 7.5 Simulation comparing M = 8 mappings using M-FSK on a power-line communications channel with background noise . 7-12 2 7.6 Graph representation of ζ7 mapping . 7-14 x List of Figures 7.7 Graph representation of all possible transpositions between symbols . 7-15 7.8 Graph representation of Construction 2 . 7-16 7.9 Graph representation of multilevel construction . 7-17 7.10 Independent transpositions for M = 8 multilevel mapping . 7-18 7.11 Symbol 1 path for M2(8, 8, 0) mapping . 7-19 7.12 Symbol paths for M2(8, 8, 0) mapping . 7-20 7.13.