Constructing Snake-in-the-box Codes and Families of such Codes Covering the Hypercube Proefschrift Ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema voorzitter van het College voor Promoties, in het openbaar te verdedigen op maandag 15 januari 2007 om 10.00 uur door Loeky Haryanto Master of Science in Mathematics, University of Florida, Master of Arts in Teaching, University of Florida, Gainesville, Florida-United States geboren te Solo, Central Java-Indonesia. Dit proefschrift is goedgekeurd door de promotor: Prof. dr. A.J. van Zanten Samenstelling promotiecommissie: Rector Magnificus, voorzitter Prof.dr. A. J. van Zanten, Universiteit Maastricht, promotor Prof. dr. S. M. Dodunekov, Bulgarian Academy of Sciences, Bulgarije Prof. dr. V. A. Zinoviev, Russian Academy of Sciences, Rusland Prof. dr. F. I. Solov’jeva, State University Novosibirsk, Rusland Prof. dr. H. C. A. van Tilborg, Technische Universiteit Eindhoven Prof. dr. C. Roos, Technische Universiteit Delft Prof. dr. C. Witteveen, Technische Universiteit Delft Prof. dr. ir. J. Biemond (reservelid) Technische Universiteit Delft Constructing Snake-in-the-box Codes and Families of such Codes Covering the Hypercube Dissertation at Delft University of Technology Copyright © 2007 by Loeky Haryanto This work has been carried out with financial support from the Royal Netherlands Academy of Arts and Sciences (KNAW) under the framework of the Scientific Programme Indonesia - Netherlands (SPIN). ISBN: 90-8559-265-8 Keywords: Standard Gray codes, symmetric transition sequence, snake-in-the-box codes (snakes), minimum-weight basis, fixed-position property, ordered basis of linear code, Reed- Muller codes, p-cover of hypercube Qn by disjoint snakes, invariance translation group. Cover: Two (vertex-)disjoint snake-in-the-box codes (snakes) cover the hypercube Q4. To my wife and children: Ogi (Oktafien Somalinggi), Bagas (Bagaskara Galois Somalinggi), and Laras (Tiara Monita Larasati). Delft, the Netherlands, January 2007 iii Acknowledgments This dissertation could not have been written without Prof. dr. A. J. van Zanten who introduced me further into the theory of algebraic coding, in particular into this special topic of ordered codes. My special thanks go to the Royal Netherlands Academy of Arts and Sciences (KNAW) for financially supporting this research, and to the Applied Mathematics Department (DIAM), TUDelft, for its hospitality and for providing me with academic facilities during the last two and half years of my work at TUDelft. My deep appreciation also goes to Mr. Paul Althuis at CICAT and to his project co- ordinators (Franca Post, Manon Post and Rene Tamboer) for their helpful administrative assistance, despite of the fact that there are so many students and Ph.D candidates to deal with. I also very much appreciate all my colleagues at TUDelft, especially my roommates Julius and Niels who always entertain me with their music, seemingly from the same song album. Neither will I forget my (former) roommates Berd and Sander, with whom I shared the same office at the 10-th floor before I had to move to the floor of DIAM. With respect to my own research, my special appreciation goes to I Nengah Suparta, the only colleague I knew at the TUDelft who was working in the area of finite ordered codes. This work is also for the memory of my late father, Lukito, who was so proud being educated in the Dutch educational system adapted in Indonesia. Unfortunately, he passed away during my stay in the Netherlands, while I was the only one of his sons and daughters who was not able to be with him in his last minutes. I am also very grateful to my late father-in-law (Somalinggi) and to my mother-in-law (Mathilda) who always prays for my success, just like my mother (Soehartati) uses to do. Last but not least, I am very grateful to my wife Oktafien, who keeps supporting me and who convinced me to be determined, and not to be worried by the absence of her and of our children (Bagas, 10, and Laras, 9) during my stay in the Netherlands, though their absence is the only thing that brings about feelings of loneliness. v Table of Contents Acknowledgments...................................................................................................... v Table of Contents......................................................................................................vii 1. Introduction ............................................................................................................1 1.1 The Main Objectives..................................................................................................1 1.2 Outline of the Next Sections......................................................................................3 2. Gray Codes and Circuit Codes...............................................................................6 2.1 The Standard Gray Code G(n) ...................................................................................6 2.2 Index Problem of G(n)...............................................................................................7 2.3 Properties of the Standard Gray Code G(n).............................................................10 2.4 DP-Codes, <m, n>- Codes and Snakes...................................................................19 3. A Construction of Snake-in-the-box Codes ..........................................................22 3.1. Snakes Based on Necklaces....................................................................................22 3.2. Generalized Necklaces............................................................................................31 3.3. A More General Construction.................................................................................32 3.4. The Structures of 2-Necklaces in Q2p, p > 2 prime.................................................41 4. Snakes Based on a Linear Code..........................................................................45 4.1 Outlines of the Construction ....................................................................................45 4.2 Necessary and Sufficient Conditions.......................................................................47 4.3 The Index Problem of a Snake in Qn .......................................................................56 5. Snakes in the Hypercubes Qn. for 3 < n ≤ 16. ......................................................59 5.1 Euclidean Geometries ..............................................................................................59 5.2 Reed-Muller Codes ..................................................................................................62 5.3. Parallel Systems in EG(3,2) and EG(4,2) ...............................................................66 5.4. Snakes Embedded in EG(3,2).................................................................................68 5.5 Snakes Embedded in EG(4, 2).................................................................................71 5.6 A Sufficient Condition for a Block List to Generate a Snake..................................79 6. Translations of Snakes.........................................................................................81 6.1 Conditions for Disjoint Translated Snakes ..............................................................81 6.2 Special Cases ...........................................................................................................87 6.3 Disjoint Snakes Based on Lexicographically Ordered Lists....................................90 7. Covers and Near-Covers of Qn, for 2 ≤ n ≤ 16.....................................................94 7.1 Symmetric Covers of Qn, for 2 ≤ n ≤ 15..................................................................94 7.2 Near-Covers of Q16 ..................................................................................................98 7.3 A Symmetric 8-Cover of Q16 .................................................................................105 vii 8. More about Covers of Hypercubes.....................................................................109 8.1 Covers of Qn and the Griesmer Bound ..................................................................109 8.2 Covers of Qn and Gray Codes................................................................................112 8.3 Special Bases for R(m − 2, m) and Covers of Qn ...................................................120 Appendices ............................................................................................................129 Appendix A..................................................................................................................129 Appendix B ..................................................................................................................132 Appendix C ..................................................................................................................136 Appendix D..................................................................................................................138 List of References ..................................................................................................139 Summary................................................................................................................143 Samenvatting .........................................................................................................144 Curriculum Vitae.....................................................................................................145