University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Dissertations, Theses, and Student Research Papers Mathematics, Department of in Mathematics 3-2018 Design and Analysis of Graph-based Codes Using Algebraic Lifts and Decoding Networks Allison Beemer University of Nebraska - Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/mathstudent Part of the Discrete Mathematics and Combinatorics Commons, and the Other Mathematics Commons Beemer, Allison, "Design and Analysis of Graph-based Codes Using Algebraic Lifts nda Decoding Networks" (2018). Dissertations, Theses, and Student Research Papers in Mathematics. 87. https://digitalcommons.unl.edu/mathstudent/87 This Article is brought to you for free and open access by the Mathematics, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Dissertations, Theses, and Student Research Papers in Mathematics by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. DESIGN AND ANALYSIS OF GRAPH-BASED CODES USING ALGEBRAIC LIFTS AND DECODING NETWORKS by Allison Beemer A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfilment of Requirements For the Degree of Doctor of Philosophy Major: Mathematics Under the Supervision of Professor Christine A. Kelley Lincoln, Nebraska March, 2018 DESIGN AND ANALYSIS OF GRAPH-BASED CODES USING ALGEBRAIC LIFTS AND DECODING NETWORKS Allison Beemer, Ph.D. University of Nebraska, 2018 Adviser: Christine A. Kelley Error-correcting codes seek to address the problem of transmitting information effi- ciently and reliably across noisy channels. Among the most competitive codes developed in the last 70 years are low-density parity-check (LDPC) codes, a class of codes whose structure may be represented by sparse bipartite graphs. In addition to having the potential to be capacity-approaching, LDPC codes offer the significant practical advantage of low- complexity graph-based decoding algorithms. Graphical substructures called trapping sets, absorbing sets, and stopping sets characterize failure of these algorithms at high signal-to- noise ratios. This dissertation focuses on code design for and analysis of iterative graph-based message- passing decoders. The main contributions of this work include the following: the unifica- tion of spatially-coupled LDPC (SC-LDPC) code constructions under a single algebraic graph lift framework and the analysis of SC-LDPC code construction techniques from the perspective of removing harmful trapping and absorbing sets; analysis of the stopping and absorbing set parameters of hypergraph codes and finite geometry LDPC (FG-LDPC) codes; the introduction of multidimensional decoding networks that encode the behavior of hard-decision message-passing decoders; and the presentation of a novel Iteration Search Algorithm, a list decoder designed to improve the performance of hard-decision decoders. iii ACKNOWLEDGMENTS First and foremost, thank you to my adviser, Christine Kelley, for her unwavering support and good humor. Her advice has been an invaluable resource, and I truly cannot thank her enough for her dedication to my work and mathematical upbringing. Thank you to my readers, Jamie Radcliffe and Judy Walker, and to Mark Brittenham, Myra Cohen, and Khalid Sayood for showing interest in my work and generously con- tributing their time to be on my supervisory committee. The Whitman College math department has my gratitude for challenging me and en- couraging me to continue on in mathematics. Deserving of particular thanks are: Barry Balof, for convincing me to be a math major, Russ Gordon, for an analysis course that got me through a qualifying exam, David Guichard, for teaching me the value of productive struggle, and Pat Keef, for his enthusiastic approach to teaching mathematics. Endless thanks are due to everyone who has made Nebraska feel like home. Among these (numerous) people are the brilliant officewomen of Avery 232, who have added a whole bunch of brightness to my life. Thank you also to family and friends who have supported me from near and far. I appreciate you all so much. Finally, thank you to Eric Canton, for an unrelenting belief in my abilities and ambi- tions, for knowing how to make me laugh at the unlikeliest of times, and for brewing me a lot of coffee. iv Table of Contents 1 Introduction 1 2 Preliminaries 4 2.1 Low-density parity-check codes . 6 2.1.1 Protograph LDPC codes . 9 2.2 Iterative decoders . 11 2.3 Trapping, stopping, and absorbing sets . 13 3 Spatially-coupled LDPC Codes 17 3.1 Construction & decoding of SC-LDPC codes . 18 3.1.1 Array-based SC-LDPC codes . 20 3.1.2 Windowed decoding . 22 3.2 Trapping set removal algorithm . 24 3.2.1 Ranking trapping sets . 27 3.2.2 Simulation results . 29 3.2.3 Trapping sets & the windowed decoder . 30 3.3 Algebraic graph lift framework for SC-LDPC code construction . 31 3.3.1 Combining edge-spreading and the terminal lift . 38 3.3.2 Comparison of construction methods . 40 3.4 Removing absorbing sets . 41 v 4 Bounds on Stopping and Absorbing Set Sizes 47 4.1 Finite geometry LDPC codes . 47 4.1.1 Smallest absorbing sets for EG0 and PG using k-caps . 52 4.1.2 Absorbing set parameters and the tree bound . 53 4.2 Hypergraph codes . 59 4.2.1 Bounds on regular hypergraph codes . 62 4.2.2 Algebraic lifts of hypergraphs . 66 5 Multidimensional Decoding Networks 71 5.1 Multidimensional network framework . 72 5.1.1 Trapping set characterization . 75 5.2 Representations yielding transitivity . 79 5.2.1 Applications . 84 5.3 Periodic decoders and the Iteration Search Algorithm . 90 6 Conclusions 100 Bibliography 101 1 Chapter 1 Introduction Mathematical coding theory addresses the problem of transmitting information efficiently and reliably across noisy channels; that is, finding methods of encoding information with the minimum redundancy required to protect against noise during transmission. In 1948, Claude Shannon pioneered the field by proving that for every channel across which one might wish to send information, there exist methods of encoding and decoding that are both arbitrarily efficient and reliable [1]. In particular, every channel has a capacity, and there exist codes and decoders for every rate below capacity such the probability of error after decoding is as small as desired. Since this revelation, coding theorists have worked to find code ensembles and decoders satisfying these conditions. Among the most competitive codes developed in the last 70 years are low-density parity-check (LDPC) codes, a class of codes whose structure may be represented by sparse bipartite graphs, or, equivalently, sparse parity-check matrices [2,3]. LDPC codes were first introduced by Gallager in his 1962 Ph.D. dissertation [2], and gained renewed interest in the coding community with the advent of turbo codes in 1993 [4]. Graph-based codes may now be found in a variety of communications applications, ranging from video streaming to data storage. Among their strengths, LDPC codes offer the significant practical advantage of low- complexity graph-based decoding algorithms. This dissertation focuses on code design for 2 iterative graph-based message-passing decoders. Graphical substructures called trapping sets characterize failure of message-passing decoding algorithms for channels with low error probability – that is, high signal-to-noise ratios (SNR) – creating what is called the error floor region of bit error rate (BER) curves. We examine trapping sets – as well as the related substructures of absorbing sets and stopping sets – and methods for their removal with reference to a variety of code families, including spatially-coupled LDPC, finite geometry LDPC, and hypergraph codes. Finally, we introduce a multidimensional network framework for the general analysis of iterative decoders. Regular LDPC codes were shown to be asymptotically good in Gallager’s original work [2]; in [5], irregular LDPC codes were optimized to be capacity-achieving. More recently, LDPC codes have been used to construct a class of codes called spatially-coupled LDPC (SC-LDPC) codes, which have been shown to exhibit threshold saturation, resulting in capacity-approaching, asymptotically good codes [6–8]. We show that the eponymous coupling step of the construction process for these codes may be exploited to remove harm- ful trapping sets in the resulting code. Furthermore, we show that the variety of existing construction methods for SC-LDPC codes may be unified using the language of algebraic graph lifts, allowing for a more systematic removal of harmful absorbing sets as well as simplified analysis of construction techniques. The inherent structure of finite geometry LDPC (FG-LDPC) codes [9] and codes con- structed from hypergraphs [10] allows us to determine harmful stopping and absorbing set parameters for these classes of codes. We show that FG-LDPC codes constructed from the line-point incidence matrices of finite Euclidean and projective planes have smallest ab- sorbing sets whose parameters meet the lower bounds shown in [11], and that these smallest absorbing sets are elementary. We then present new results on the number of erasures cor- rectable in regular hypergraph codes, and give a proof of the existence of infinite sequences of regular hypergraphs with expansion-like properties by introducing a construction for 3 hypergraph lifts. Finally, we present a multidimensional decoding network framework that encodes the behavior of any hard-decision message-passing decoder for a code with a given graph rep- resentation. Notably, the trapping sets of a code may be easily seen using the decoding network of the code and chosen decoder. Decoding networks with a particular transitivity property allow for a simplified application of results; we show that every code with dimen- sion at most two less than its block length has a transitive representation under the Gallager A decoding algorithm. We also introduce the decoding diameter, aperiodic length, and pe- riod of a decoding network as parameters containing essential information about the inter- action of the code and decoder.