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Coding Theory Via Groebner Bases Coding Theory via Groebner Bases Vom Promotionsausschuss der Technischen Universitat¨ Hamburg-Harburg zur Erlangung des akademischen Grades Doktorin der Naturwissenschaften genehmigte Dissertation von Mehwish Saleemi aus Islamabad 2012 1. Gutachter: Prof. Dr. Karl-Heinz Zimmermann Institute fur¨ Rechnertechnologie, Technische Universitat¨ Hamburg-Harburg 2. Gutachter: Prof. Dr. Rudolf Scharlau Fakultat¨ fur¨ Mathematik, Technische Universitat¨ Dortmund Tag der mundlichen¨ Prufung:¨ 14.02.2012 Vorsitzender des Prufungsausschusses:¨ Prof. Dr. Dieter Gollmann Institut fur¨ Sicherheit in verteilten Anwendungen, Technische Universitat¨ Hamburg- Harburg Abstract Coding theory plays an important role in efficient transmission of data over noisy communication channels. It consists of two steps; the first step is to encode the data to reduce its sensitivity to noise during transmission, and the second step is to decode the received data by detecting and correcting the noise induced errors. In this thesis an algebraic approach is used to develop efficient encoding and decod- ing algorithms for a very commonly used class of linear codes, the Reed-Muller codes and the Golay codes. To develop the approach first the algebraic structure of linear codes is explored. For this, the reduced Groebner basis for a class of ideals in commutative poly- nomial rings is constructed. The extension of these ideals to a residue class ring enabled us to find the parameters of the corresponding codes. It is found that the corresponding codes contains the primitive Reed-Muller codes. The added advan- tage of this approach is that, once these Groebner bases are constructed a standard procedure can be used to develop encoding and decoding processes. A binomial ideal, defined as a sum of toric ideal and a prime ideal over some arbitrary field, is explored. It is shown that this ideal is equal to a binomial ideal over a prime field. Purpose of proving this equivalence is to study binary codes associated to this ideal. Minimal generators and Groebner basis found for this ideal showed that the situation is quite closely related to the toric case. The investigation of uni- versal Groebner basis, Graver basis and circuits for this ideal revealed that they have the same relationship among them which is true in general for toric ideals. Each linear code can be described as a binomial ideal defined above. Since the reduced Groebner basis for any ideal plays a vital role in describing encoding and decoding processes for the corresponding codes, a natural reduced Groebner basis for this ideal is proposed for any general term order. In fact, if a generator ma- trix is given for any code, by constructing the corresponding particular binomial ideal, one can immediately describe the reduced Groebner basis. Information po- sitions and parity check positions are then given by standard and non-standard monomials for the ideal. A systematic encoding algorithm for such codes is ex- plained in terms of remainders of the information word computed with respect to the reduced Groebner basis. Furthermore, the binary and ternary Golay codes are studied algebraically in terms of the binomial ideal. Finally, a presentation of the binomial ideal of a linear code in terms of its syzygy modules is provided and the corresponding finite free resolution has been described. Acknowledgements My deepest gratitude is to my supervisor Professor Karl-Heinz Zimmermann. His generous advice, inspiring guidence, encouragement and benevolence helped me tremendously through out my work. I am extremely grateful to him for his help and support, without which it would have been impossible for me to complete this thesis. I feel very priviledged indeed to have been one of his students. My warm thanks are due to my colleague, Dr. Svetlana Torgasin, not only for generously giving her time to help me with latex but also for being there whenever I needed her. I am gratefull to all my colleagues and staff for their support and providing friendly atmosphere. Special thanks go to Mr. Stefan Just for providing technical help. Finally, I wish to express my love and gratitude to all my family, in-laws and friends for their encouragement and emotional support. I want to express my deep appreciation and love for my parents for their prayers, unconditional love and endless support in all my pursuits. I must express my gratitude to Saulat, my husband, for his continued support, encouragement and great sense of humor. The forbearance shown by my children has been my greatest asset in concluding this study. To them, I am eternally grateful. Contents 1 Introduction 3 1.1 The Problem ............................. 3 1.2 Previous Work ............................ 4 1.3 Contribution of the Thesis ..................... 5 1.4 Organization ............................. 6 2 Polynomial Algebra 7 2.1 Monomials ............................. 7 2.2 Groebner Bases ........................... 11 2.3 Classification of Groebner bases .................. 12 2.3.1 The Reduction Process ................... 12 2.4 Computing Groebner Bases ..................... 12 2.5 Binomial Ideal ........................... 14 2.6 Groebner Bases for Modules .................... 15 2.7 Syzygies and Finite Free Resolution ................ 19 3 Algebraic Coding Theory 23 3.1 Basic Coding Theory ........................ 23 3.2 Finite Fields ............................. 26 2 CONTENTS 3.3 Linear Codes ............................ 28 3.4 Syndrome Decoding ........................ 29 3.4.1 Decoding Linear Codes ................... 30 3.5 Cyclic Code ............................. 31 3.6 Ideals as Linear Codes ....................... 32 3.7 Families of Codes .......................... 34 3.7.1 Hamming Codes ...................... 34 3.7.2 Reed Muller Codes ..................... 35 3.7.3 Golay Codes ........................ 37 4 Variants of Reed Muller Codes 39 4.1 Introduction ............................. 39 4.2 Groebner Basis Construction .................... 39 4.3 Encoding Linear Codes using Groebner Bases ........... 44 4.4 Variants of Primitive Reed-Muller Codes .............. 46 4.5 Variants of Primitive Reed-Muller Codes with Designated Distance 48 5 Linear Codes as Binomial Ideals 51 5.1 Introduction ............................. 51 5.2 Groebner Basis of the Ideal IA;P ................... 51 5.3 Ideal Bases ............................. 57 5.4 Reduced Groebner Basis of IC ................... 62 5.4.1 Application to Golay Codes ................ 63 5.5 Decomposition of the Ideal IC ................... 65 5.6 Affine Varieties ........................... 67 5.7 Encoding .............................. 69 CONTENTS 3 6 Syzygies 71 6.1 Introduction ............................. 71 6.2 Syzygies and Free Resolutions of Linear Codes .......... 71 7 Conclusions and Future Directions 77 Bibliography 79 Notation d(x; y) Hamming distance between x and y d Minimum distance of a code s Syndrome of a received vector [n; k; d] A linear code with lenght n, dimension k and minimum distance d K Field K[x] Polynomial ring in n variables over K ≻ Term ordering lt( f ) Leading term of a polynomial f lt(I) Leading ideal of an ideal I S( f; g) S-polynomial of f and g G Groebner basis of an ideal V(I) Variety of an ideal p I Radical of an ideal I Chapter 1 Introduction 1.1 The Problem The problem of robust information transmission is one of the problems that is encountered in modern data transmission due to the presence of hostile enviroment during transmission, which thus results into the distorted information. In order to combat this situation error-correcting codes are introduced [42]. A fundamental problem is to send a message across a noisy channel with a maximum possible reliability. Error correcting codes are used to correct messages when they are transmitted through noisy channels(Figure 1.1). Message Source Receiver Source Encoder Channel Source Decoder Figure 1.1: A basic communication model The main features of coding theory are: • efficient encoding of messages, • easy transmission of encoded messages, 4 1. Introduction • fast and reliable decoding of recieved messages, • transmission of large number of messages per unit of time. 1.2 Previous Work The last several years have witnessed major theoretical advances in coding the- ory resulting in a new coding concepts such as algebraic codes, codes on graphs etc. With the publication of C. Shannon’s [54] seminal paper the whole new sub- ject of coding theory was inaugurated. Later, R.W. Hamming [30] proposed a first method of encoding data so that errors can not only be detected but can be corrected too. With this coding theory started to develop along two main direc- tions: probabailistic coding theory and algebraic coding theory. The basic idea is to protect a message by adding some redundant information, thus in case even if the message is corrupted, enough redundancy can help in recovering the message completely. The main objects of study in algebraic coding are codes that are lin- ear subspaces of finite-dimensional vector spaces over a finite field. In particular, research was mainly devoted to cyclic codes that form a class of linear codes al- lowing easier determination. The most important cyclic codes, BCH codes and Reed Solomon codes [8, 45, 32] were discovered between 1958 and 1960. The theory of algebraic codes indicates the fact that by adding more algebraic struc- ture to the system, better descriptions can be obtained. A perfect illustration of this fact is the work of S. D. Berman [5], who discovered that the Reed-Muller F codes
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