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Applications of Finite Geometries to Designs and Codes Michigan Technological University Digital Commons @ Michigan Tech Dissertations, Master's Theses and Master's Dissertations, Master's Theses and Master's Reports - Open Reports 2011 Applications of finite geometries ot designs and codes David C. Clark Michigan Technological University Follow this and additional works at: https://digitalcommons.mtu.edu/etds Part of the Mathematics Commons Copyright 2011 David C. Clark Recommended Citation Clark, David C., "Applications of finite geometries ot designs and codes", Dissertation, Michigan Technological University, 2011. https://doi.org/10.37099/mtu.dc.etds/199 Follow this and additional works at: https://digitalcommons.mtu.edu/etds Part of the Mathematics Commons APPLICATIONS OF FINITE GEOMETRIES TO DESIGNS AND CODES By David C. Clark A DISSERTATION Submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY (Mathematical Sciences) MICHIGAN TECHNOLOGICAL UNIVERSITY 2011 c 2011 David C. Clark This dissertation, "Applications of Finite Geometries to Designs and Codes," is hereby approved in partial fulfillment of the requirements for the Degree of DOC- TOR OF PHILOSOPHY IN MATHEMATICAL SCIENCES. Department of Mathematical Sciences Signatures: Dissertation Advisor Dr. Vladimir Tonchev Committee Member Dr. Donald Kreher Committee Member Dr. Stefaan De Winter Committee Member Dr. Steven Seidel Department Chair Dr. Mark Gockenbach Date Contents List of Figures ..................................... ix List of Tables ...................................... xi Preface .........................................xiii Acknowledgments ................................... xv Abstract .........................................xvii 1 Designs, codes, and finite geometries ....................... 1 1.1 Introduction .................................. 1 1.1.1 Designs ................................ 2 1.1.2 Error-correcting codes ........................ 4 1.1.3 The codes of a design and p-ranks .................. 5 1.2 Finite geometry designs and codes ...................... 6 1.2.1 Projective Geometry Designs ..................... 6 v 1.2.2 Affine Geometry Designs ....................... 7 1.2.3 The finite geometry codes and their duals .............. 9 1.2.4 Decoding schemes .......................... 10 1.3 Summary and contributions .......................... 12 2 Affine geometry designs, polarities, and Hamada’s conjecture ......... 13 2.1 Ranks of incidence matrices and Hamada’s conjecture ............ 13 2.1.1 Ranks of Finite Geometry Designs .................. 14 2.1.2 Hamada’s conjecture ......................... 18 2.1.3 Proved cases ............................. 19 2.1.4 Counterexamples ........................... 24 2.2 Introduction .................................. 26 2.3 Polarity designs from AGd+1(2d + 1,q) for q > 2 .............. 35 3 Multi-step majority logic decoding and the modified finite geometry designs . 39 3.1 Introduction .................................. 39 3.2 Majority logic decoding ............................ 40 3.3 Decoding finite geometry codes ........................ 42 3.4 Decoding the modified finite geometry codes ................. 44 3.4.1 Modified projective geometry designs ................ 44 vi 3.4.2 Modified affine geometry designs .................. 48 3.5 Minimum distances .............................. 50 4 Nonbinary quantum codes derived from finite geometries ........... 55 4.1 Introduction .................................. 55 4.2 Quantum codes from projective geometry .................. 57 4.3 Quantum codes from affine geometry ..................... 61 4.4 Acknowledgments ............................... 63 5 Entanglement-assisted quantum low-density parity-check codes ....... 65 5.1 Introduction .................................. 65 5.2 Combinatorial entanglement-assisted quantum LDPC codes ......... 67 5.2.1 Preliminaries ............................. 68 5.2.2 General combinatorial constructions ................. 71 5.3 Finite geometry codes ............................. 81 5.3.1 Projective geometry codes ...................... 82 5.3.1.1 Point-by-block (Type II) Projective geometry codes .... 83 5.3.1.2 Block-by-point (Type I) Projective geometry codes .... 88 5.3.2 Affine geometry codes ........................ 90 5.3.2.1 Point-by-block (Type II) Affine geometry codes ..... 91 vii 5.3.2.2 Block-by-point (Type I) Affine geometry codes ...... 93 5.3.3 Euclidean geometry codes ...................... 96 5.4 Performance .................................. 99 5.5 Conclusion ..................................104 5.A Appendix A: Existence of 2-designs .....................107 5.B Appendix B: Parameters of quantum and classical FG-LDPC codes with girth six ....................................108 6 Summary and future work ............................111 6.1 Summary ...................................111 6.2 Future work ..................................112 References .......................................115 C Copyright documentation .............................125 viii List of Figures 1.1TheFanoplane................................ 2 5.1 Performance of Type I EAQECCs ......................100 5.2 Performance of Type II EAQECCs ......................102 5.3 Performance of Type II EAQECCs ......................103 5.4 Performance of high-rate Type II EAQECCs .................104 5.5 Performance of EAQECCs obtained by deleting subdesigns from AG1(3,3). 106 ix x List of Tables c 4.1 Sample parameters of p-ary quantum codes obtained from PGt(m, p ).... 61 4.2 Sample parameters of p-ary quantum codes obtained from AGt(m,q). ... 63 5.1 Sample parameters of Type II [[n,k,d;c]] EAQECCs obtained from PG1(m,q), q even...................................... 86 5.2 Sample parameters of Type II [[n,k,d;c]] EAQECCs obtained from PG1(m,q), q odd. ..................................... 87 5.3 Summary of parameters of Type II codes obtained by deleting a Steiner spread of subdesigns isomorphic to PG1(2,2) from PG1(5,2). ....... 88 5.4 Sample parameters of Type I [[n,k,d;c]] EAQECCs obtained from PG1(m,q), q even...................................... 89 5.5 Sample parameters of Type II [[n,k,d;c]] EAQECCs obtained from AG1(m,q), q even...................................... 93 5.6 Sample parameters of Type II [[n,k,d;c]] EAQECCs obtained from AG1(m,q), q odd. ..................................... 93 5.7 Summary of parameters of Type II codes obtained by deleting a Steiner spread of subdesigns isomorphic to AG1(2,4) from AG1(3,4). ....... 94 5.8 Sample parameters of Type I [[n,k,d;c]] EAQECCs obtained from AG1(m,q), q even...................................... 95 xi 5.9 Sample parameters of Type I [[n,k,d;c]] EAQECCs obtained from EG1(2,q), q even...................................... 97 5.10 Sample parameters of Type II [[n,k,d;c]] EAQECCs obtained from EG1(m,q), q even...................................... 99 5.11 Sample parameters of Type II [[n,k,d;c]] EAQECCs obtained from EG1(m,q), q odd. .....................................100 5.12 Rates of EAQECCs obtained from finite geometries. .............105 5.13 Summary of parameters of Type II EAQECCs obtained by deleting subde- signs from AG1(3,3). .............................105 5.14 Parameters of entanglement-assisted quantum LDPC codes from finite ge- ometries. ....................................109 5.15 Parameters of classical FG-LDPC codes. ...................110 xii Preface Several chapters in this dissertation are the result of collaborative work, and have been published in or submitted to refereed journals. Each paper is written according to the style of the journal in which it was published. The introduction to each paper has been left in place, even though this may repeat definitions which are given in Chapter 1. Several papers have been modified by adding an extended introductory section concerning the history of the paper’s topic, as well as minor editorial changes. Documentation of permission to reprint each article is documented in Appendix C. • Chapter 2: Affine geometry designs, polarities, and Hamada’s conjecture. This is joint work with D. Jungnickel and V. D. Tonchev. The topic was suggested by V. D. Tonchev. The author performed the research and wrote the paper, with guidance and editing provided by D. Jungnickel and V. D. Tonchev. It was previously published in the Journal of Combinatorial Theory, Series A. • Chapter 4: Nonbinary quantum codes derived from finite geometries. This is joint work with V. D. Tonchev. The topic was suggested by V. D. Tonchev. The author performed the research and wrote the paper, with guidance and editing provided by V. D. Tonchev. It was previously published in the journal Finite Fields and their Applications. • Chapter 5: Entanglement-assisted quantum low-density parity-check codes. This is joint work with Y. Fujiwara, P. Vandendriessche, M. De Boeck, and V. D. Tonchev. The topic was suggested by Y. Fujiwara. The author performed the research and writing for Section III, except for two theorems concerning minimum distance. The author also performed the simulations and created the plots and tables in Section IV. The author is also responsible for portions of the writing, editing, and figures in all other sections. It was previously published in the journal Physical Review A. xiii xiv Acknowledgments My sincere thanks go to my advisor, Vladimir Tonchev. Throughout my graduate career, Professor Tonchev has provided me with an amazing range of opportunities. His encour- agement and ideas have directed the course of my research, and I would not be where I am today without
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