UC San Diego UC San Diego Electronic Theses and Dissertations Title Algebraic list-decoding of error-correcting codes Permalink https://escholarship.org/uc/item/68q346tn Author Parvaresh, Farzad Publication Date 2007 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA, SAN DIEGO Algebraic List-Decoding of Error-Correcting Codes A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Electrical Engineering (Communications Theory and Systems) by Farzad Parvaresh Committee in charge: Professor Alexander Vardy, Chair Professor Ilya Dumer Professor Alon Orlitsky Professor Paul H. Siegel Professor Jack K. Wolf 2007 Copyright Farzad Parvaresh, 2007 All rights reserved. The dissertation of Farzad Parvaresh is approved, and it is acceptable in quality and form for publication on microfilm: Chair University of California, San Diego 2007 iii To my parents iv TABLE OF CONTENTS Signature Page . iii Dedication . iv Table of Contents . v List of Figures . vii Acknowledgements . viii Vita and Publications . xi Abstract of the Dissertation . xii Chapter 1 Introduction . 1 1.1 Contributions . 5 1.2 Waiting for a bus in front of a pub . 8 1.2.1 Reed-Solomon codes . 9 1.2.2 Guruswami-Sudan algorithm . 10 1.2.3 Multivariate interpolation decoding algorithm . 12 1.3 Waiting for a bus on an April Fool’s day . 15 1.4 A new kind of Tic-Tac-Toe with multiplicity . 18 Chapter 2 Multivariate decoding of Reed-Solomon codes . 21 2.1 Introduction . 22 2.2 Preliminaries . 27 2.3 Error-correction radius of the MID algorithm . 30 2.4 Iterative multivariate interpolation . 35 2.5 Multivariate polynomial recovery . 39 2.6 The MID algorithm works . 44 2.7 A bound on the performance of the MID algorithm . 46 2.7.1 Background and notation . 47 2.7.2 Bound on the failure probability . 48 v Chapter 3 Correcting errors beyond the Guruswami-Sudan radius . 58 3.1 Introduction . 59 3.1.1 Prior work on the list decoding problem . 61 3.1.2 Our main results . 63 3.1.3 The underlying ideas . 65 3.1.4 Overview of the chapter . 66 3.2 Multivariate interpolation . 67 3.3 Simple trivariate codes and their decoding . 70 3.3.1 Code parameters and encoder mapping . 71 3.3.2 Simple trivariate interpolation decoding . 73 3.4 General trivariate codes and their decoding . 77 3.4.1 Code parameters and encoder mapping . 77 3.4.2 General trivariate interpolation decoding . 79 3.5 Extension to multivariate interpolation . 81 3.6 Explicit capacity achieving codes . 84 3.6.1 Folded Reed-Solomon codes . 85 3.6.2 Using trivariate interpolation for folded RS codes . 86 Chapter 4 Multiplicity assignment . 90 4.1 Introduction . 91 4.2 Background and notation . 93 4.3 Geometrical framework . 96 4.4 Second-order approximation . 102 4.4.1 Iterative solution . 104 4.4.2 Analytical solution . 106 4.5 Fixed-cost multiplicity assignment . 111 Chapter 5 Efficient interpolation via matrix-chain product . 116 5.1 Introduction . 117 5.2 Interpolation as a matrix-chain product . 119 5.3 Optimal multiplication order . 121 5.4 Suboptimal dynamic-programming algorithm . 124 5.5 Optimization over the interpolation polynomial . 125 Chapter 6 Conclusion and open problems . 129 References . 134 vi LIST OF FIGURES Figure 1.1 Schematic of a generic communication system . 2 Figure 1.2 q-ary symmetric channel . 8 Figure 1.3 Monomials of weighted degree smaller than ¢ . 12 Figure 1.4 Error-correction radius of trivariate interpolation decoding 13 Figure 1.5 Error-correction radius of different codes in the worst-case 16 Figure 1.6 Tic-Tac-Toe with multiplicity . 19 Figure 2.1 Error-correction radius of trivariate interpolation decoding 24 Figure 2.2 Comparison of algorithms that correct synchronized errors 25 Figure 2.3 The union of cubes S and the pyramid P . 32 Figure 2.4 Deltaset of the ideal IP .................... 41 Figure 2.5 Deriving the bound for ¢2 ................... 42 Figure 2.6 Bound on decoding with two Grobner¨ basis . 43 Figure 2.7 Performance of MID of C256(15, 11) . 45 Figure 2.8 Performance of MID of C256(15, 7) . 46 Figure 2.9 Counting the number of curves . 50 Figure 2.10 Comparing upper and lower bounds on the weighted degree 54 ¤ Figure 2.11 Deriving the bound on Wdeg G (X, Y, Z) . 57 Figure 3.1 Unfolding a folded Reed-Solomon code . 86 Figure 4.1 Score distributions for a fixed n . 92 Figure 4.2 Geometrical framework . 100 Figure 4.3 Performance of C16(15, 11) . 101 Figure 4.4 Performance of decoding C(468, 420) over AWGN channel 106 Figure 4.5 Performance of decoding C(255, 239) over AWGN channel 107 Figure 4.6 Geometrical framework; The cone and tangent rays . 113 Figure 5.1 Cost of multiplying a 4 £ 4 polynomial matrix-chain . 128 Figure 5.2 Cost of multiplying an 8 £ 8 polynomial matrix-chain . 128 vii ACKNOWLEDGEMENTS First and foremost, I like to thank my advisor, Alexander Vardy. Now that I look back to the time I got admission from UCSD, and started working with Alex, it all looks like a miracle to me. I truly believe, during the past five years, I have been evolved as a scientist and a person, and more than any other person, Alex was influential in this change. This work would not be possible with the very friendly and supportive research environment that he provided for me and his constant attention and believe on what I was doing for my research. I feel very lucky to have had Alex Vardy as my advisor and look forward to learn and work with him in future. I like to thank my dissertation committee members. I am very grateful to Paul Siegel. I learned a lot about coding theory from his classes and enjoyed attending STAR group meetings. I am grateful to Alon Orlitsky. Alon’s class on information theory had a huge impact on attracting me toward coding and information theory. Also, I could not imagine doing my Ph.D. at UCSD without his help with my visa. Jack Wolf has been a role model for me. I am hoping that some day I would be able to explain complex subjects in simple terms like him. I am grateful to Ilya Dumer. Ilya’s insight that the bounds of Chapter 2 are optimal in a certain sense makes the results of Chapter 2 much stronger. And, although he had to drive from Riverside to San Diego each time for my qualifi- cation exam and defense, he kindly agreed to be in my defense committee. I had a chance to do an internship at Bell laboratories under the supervision of Alexei Ashikhmin. I learned a lot about quantum codes from Alexei during the short period that I spent at Bell Labs. I am very grateful to him for his magic introduction to tough concept of quantum codes in a language which is plausible for a coding guy. I also like to thank Gerhard Kramer, Tom Marzetta, Emina Soljanin, Adriaan van Wijngaarden and others at Bell Labs mathematical research center for making my stay at Bell Labs an enjoyable one. I am grateful to Bob McEliece for giving me a chance to visit his group at Caltech and Shuki Bruck for letting me to attend his group meetings during my stay at Caltech. I truly enjoyed the discussions in the meetings which helped viii me a lot to diversify my research. I also like to thank Ralf Koetter. Ralf’s suggestions on my research always helped me to understand the problem in a much deeper level . I hope someday I will be as comfortable as he is with algebra. I like to acknowledge Madhu Sudan and Venkat Guruswami for their invaluable comments on my research. I would also like to thank my officemates, Nandu Santi for all the enjoy- able discussions we had, Jun Ma, Navin Kashyap and Moshe schwartz. I like to thank my friends, Houman Ghajari for being a great roommate and great friend, Amin Shameli, Alireza Vakil Amirkhizi, Mohsen Bayati, Navid Ehsan, Moham- mad Farazian , Zeinab Taghavi, Hossein Taghavi, Kambiz Samadi, Amir Fara- jidana, Shafigh Shirinfard, Aydin Babakhani, Olgica Milenkovic, Narayana P. Santhanam, Mostafa Al-Khamy, Mihailo Stojnic, Ashay Dhamdhere, Mishal Al- gharabally, Sharon Aviran, Liwen Yu, Seyhan Karkulak, Mike Cheng, Joseph So- riaga, Henry Pfister, Junsheng Han, Patrick Amihood, Chaitanya Rao, Sormeh Shadbakht, Ali Vakili, Weiyu Xu, Fred´ erique´ Oggier, Haris Vikalo, Vincent Bo- hossian, Yuval Cassuto, Michael Langberg, Sarah Fargol, Edwin Soedarmadji, and many others that made my graduate life a fruitful one. I am specially grateful to my family, to my father, Ahmad, for always being there for me, paying attention to all the details of my work and giving his help and support in all aspect of my life, to my mom, Parvin, for always encouraging and complementing my work and giving me confidence on what I am doing and my sister, Maryam, which her sense of humor made my telephone calls more fun. Although, I did not have a chance to visit them during my study at UCSD, I had their love and support even from such a long distance. And, I am very grateful to my uncles, Hushang, Farideh, Parynaz, Shahram, Peyman, Ebi, Ann and my second cousins for their support and for making me feel that I am back home here in US. The results of the Chapter 2 have been presented, in part, at the 43ed Annual Allerton Conference on Communication, Control, and Computing, Parvaresh, Farzad; Vardy, Alexander, and at the 2006 International Symposium on Information Theory (ISIT), Parvaresh, Farzad; Taghavi, Mohammad; Vardy, Alexander.