University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Dissertations, Theses, and Student Research Papers Mathematics, Department of in Mathematics Spring 4-25-2014 Combinatorial and Algebraic Coding Techniques for Flash Memory Storage Kathryn A. Haymaker University of Nebraska-Lincoln, [email protected] Follow this and additional works at: http://digitalcommons.unl.edu/mathstudent Part of the Discrete Mathematics and Combinatorics Commons, and the Other Applied Mathematics Commons Haymaker, Kathryn A., "Combinatorial and Algebraic Coding Techniques for Flash Memory Storage" (2014). Dissertations, Theses, and Student Research Papers in Mathematics. 53. http://digitalcommons.unl.edu/mathstudent/53 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. COMBINATORIAL AND ALGEBRAIC CODING TECHNIQUES FOR FLASH MEMORY STORAGE by Kathryn Haymaker 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 May, 2014 COMBINATORIAL AND ALGEBRAIC CODING TECHNIQUES FOR FLASH MEMORY STORAGE Kathryn Haymaker, Ph. D. University of Nebraska, 2014 Adviser: Christine A. Kelley Error-correcting codes are used to achieve reliable and efficient transmission when storing or sending information across a noisy channel. This thesis investigates a mathematical approach to coding techniques for storage devices such as flash memory storage, although many of the resulting codes and coding schemes can be applied in other contexts. The main contributions of this work include the design of efficient codes and decoding algorithms using discrete structures such as graphs and finite geometries, and developing a variety of strategies for adapting codes to a multi-level setting. Information storage devices are prone to errors over time, and the frequency of such errors increases as the storage medium degrades. Flash memory storage tech- nology has become ubiquitous in devices that require high-density storage. In this work we discuss two methods of coding that can be used to address the eventual degradation of the memory. The first method is rewriting codes, a generalization of codes for write-once mem- ory (WOM), which can be used to prolong the lifetime of the memory. We present constructions of binary and ternary rewriting codes using the structure of finite Eu- clidean geometries. We also develop strategies for reusing binary WOM codes on multi-level cells, and we prove results on the performance of these strategies. The second method to address errors in memory storage is to use error-correcting iii codes. We present an LDPC code implementation method that is inspired by bit- error patterns in flash memory. Using this and the binary image mapping for non- binary codes, we design structured nonbinary LDPC codes for storage. We obtain performance results by analyzing the probability of decoding error and by using the graph-based structure of the codes. iv COPYRIGHT c 2014, Kathryn Haymaker v DEDICATION This dissertation is lovingly dedicated to my parents Thomas and Ann Haymaker, and in fond remembrance of Dr. Jenna Higgins. vi ACKNOWLEDGMENTS This dissertation would not have been possible without the work of Professor Christine Kelley, who has generously provided me with guidance and support since my first week at UNL. I was fortunate to have such a dedicated and insightful mentor. Working with Professor Kelley helped me grow mathematically and professionally, and I will be forever grateful for the many opportunities that she gave me. I would also like to thank the members of my doctoral committee: Professor Judy Walker and Professor Jamie Radcliffe, for providing me with valuable feedback at many points during my graduate career, and Professor Myra Cohen and Professor David Pitts, for showing consistent interest in my work. My family has given me immense support. This dissertation is dedicated to my parents, because their selfless love is beyond words. They would go to the end of the world for the sake of family, but instead they ended up repeatedly driving to Nebraska (and beyond). Thank you to my siblings Kelly and Joey, for always challenging me to improve by being better than me at many things. I am also grateful to my grandparents for providing me with a strong example of honest hard work, which has been particularly inspiring these past few years! I would like to acknowledge my officemates at UNL: Abby, Kaelly, and Molly, for being excellent friends and making our first year cozy in our three-person office, and Simone for many thoughtful conversations during my last year. Molly has also been a wonderful roommate, co-teacher, and work partner. Thank you to Ashley and Melanie, for work time and crossword time, and everything in between. Your friendship means so much to me. The UNL Department of Mathematics provides a wonderful environment for graduate students, and I have enjoyed learning and teaching alongside this group of talented individuals. I owe particular thanks to vii Amanda, Anisah, Ben, Courtney, James, Lauren, and Sara for being amazing, each in amazingly different ways. I would also like to acknowledge the other members of RAD for making our \recreational research" so much fun, even in the midst of realizing we had been scooped. Thank you to Bo Zhang and Angela Bliss for providing helpful comments throughout the past two years of the dissertation writing process. Professor Rhonda Hughes of Bryn Mawr College first inspired me to pursue math- ematics and continues to provide inspiration. I am also grateful to Professor Paul Melvin and Professor Leslie Cheng for their encouragement and support. Thank you to my dear friends from Bryn Mawr|particularly Alice, Talia, Nicole, and Jenna. Finally, I would like to thank Nathan Corwin for infusing my life with an abun- dance of humor and love. viii GRANT INFORMATION This work was supported in part by the National Security Agency under Grant Num- ber H98230-11-1-0156. The United States Government is authorized to reproduce and distribute reprints not-withstanding any copyright notation herein. This work was also supported in part by a University of Nebraska Presidential Fellowship, the United States Department of Education GAANN grant number P200A090002, and an NSF EPSCoR First Award. ix Contents Contents ix List of Figures xii List of Tables xv 1 Introduction1 2 Preliminaries5 2.1 Error-correcting codes .......................... 7 2.1.1 Hamming codes.......................... 9 2.1.2 Reed-Muller codes ........................ 10 2.1.3 LDPC codes............................ 11 2.2 Coding for flash memories........................ 15 2.2.1 Flash memory structure ..................... 15 2.2.2 WOM codes............................ 17 2.2.3 Flash Codes............................ 19 3 Write-once memory codes from finite geometries 21 3.1 Finite geometries............................. 21 3.2 The Merkx construction ......................... 23 x 3.3 WOM codes from EG(m; 2)....................... 25 3.3.1 Comparison............................ 29 3.4 Ternary flash codes from EG(m; 3) ................... 31 3.4.1 Encoding and decoding...................... 31 3.4.2 Examples ............................. 33 4 Coding methods for multi-level flash memories 38 4.1 Strategies for reusing binary WOM codes................ 39 4.1.1 Analysis of Strategies A and B ................. 42 4.2 Concatenation with WOM codes .................... 46 4.3 Generalized position modulation..................... 50 4.3.1 GPM-WOM code construction.................. 52 4.3.2 Examples and code performance................. 55 4.4 Coset encoding on multi-level cells ................... 58 4.4.1 Binary coset encoding ...................... 58 4.4.2 Construction I........................... 60 4.4.3 Construction II.......................... 63 4.4.4 Codes from Constructions I and II ............... 65 4.4.5 Error correction.......................... 68 5 Binary structured bit-interleaved LDPC codes 72 5.1 Motivation from MLC flash memory .................. 75 5.2 Bit assignments for binary regular LDPC codes ............ 76 5.2.1 Results for binary (j; k)-regular codes with Gallager A . 80 5.2.2 Results for binary (j; k)-regular codes with Gallager B . 84 5.3 More than two check node types..................... 86 5.4 Results in terms of noise variance and SNR thresholds . 88 xi 6 Nonbinary structured bit-interleaved LDPC codes 94 6.1 The binary image of a code ....................... 96 6.2 Implementing nonbinary codes in MLC flash.............. 97 6.3 Designing codes with nonbinary edge labels . 100 6.3.1 Performance of binary expanded graph decoding in terms of bi thesholds ............................. 102 6.3.2 Nonbinary performance in terms of noise variance and SNR thresholds............................. 108 6.4 Performance using nonbinary decoding . 109 7 Bounds on the covering radius of graph-based codes 119 7.1 Graph-based bound on covering radius . 120 7.2 LDPC codes from finite geometries ................... 121 7.3 Covering radius of Euclidean geometry LDPC codes . 124 7.4 Covering radius of projective geometry LDPC codes . 128 8 Conclusions 134 Bibliography