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Difference Sets from Projective Planes

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Difference Sets from Projective Planes View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Digital University Library Saxony-Anhalt Difference Sets From Projective Planes Dissertation zu Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) von M. Sc. Yue Zhou geb. am 16.07.1984 in Shanxi genehmigt durch die Fakultät für Mathematik der Otto-von-Guericke-Universität Magdeburg. Gutachter: Prof. Dr. Alexander Pott Prof. Dr. Michel Lavrauw eingereicht am: 28.03.2013 Verteidigung am: 28.06.2013 Zusammenfassung Wir befassen uns in dieser Dissertation mit mehreren Typen verallgemeinerter Differenzmengen in abelschen Gruppen. Zunächst konzentrieren wir uns auf (q, q, q, 1)-relative Differenzmengen mit ungeradem q. Diese sind äquivalent zu planaren Funktionen über Fq. Die mei- sten bekannten Beispiele führen zu kommutativen Halbkörpern. Eines unserer Hauptresultate ist die Konstruktion einer neuen Familie von Halbkörpern un- gerader Charakteristik. Wir bestimmen ihre linken, rechten und mittleren Nu- klei. Wir zeigen, dass dieser Halbkörper für bestimmte Parameter auf zwei nicht äquivalente planare Funktionen führen. Durch Anwendung der Charakter-Methode auf planare Funktionen über Fq mit q ungerade zeigen wir einige überraschende Beziehungen zwischen pla- naren Funktionen über Fpm und planaren Funktionen über Fp2m . Wir untersu- chen Projektionen und Hochhebungen von planaren Funktionen, wobei wir auch einen Ansatz der Kodierungstheorie ausnutzen. Wir geben einige Resultate com- putergestützter Rechnungen über die “Switchings” planarer Funktionen von F3n für n = 3, 4, 5, 6 an. n n n n Danach untersuchen wir (2 , 2 , 2 , 1)-relative Differenzmengen in C4 rela- n tiv zu C2 . Dabei werden zwei Darstellungen eingeführt, von denen eine das Analogon klassischer planarer Funktionen ist, welche auf Fq mit q ungerade de- finiert sind. Genauer gesagt führt eine Funktion f auf F2n genau dann zu einer (2n, 2n, 2n, 1)-Differenzmenge, wenn die Abbildung x 7! f (x + a) + f (x) + f (a) + xa für beliebige a 6= 0 eine Permutation auf F2n ist. Solche f nennen wir ebenfalls planare Funktionen. Wir zeigen, dass eine planare Funktion f auf F2n genau dann als Dembowski-Ostrom-Polynom geschrieben werden kann, wenn x ∗ a := f (x + a) + f (x) + f (a) + xa die Multiplikation auf einem Halbkörper n der Ordnung 2 definiert. Für den Fall, dass f eine Abbildung auf F2n mit den Eigenschaften f (0) = 0 und Im( f ) = f0, xg mit x 6= 0 ist, beweisen wir, dass f genau dann eine planare Abbildung ist, wenn f (x + y) = f (x) + f (y) für alle x, y 2 F2n gilt. Wir betrachten auch Projektionen von planaren Funktionen, die wir “shifted-bent” Funktionen nennen. Wir zeigen einen interessante Zusam- n menhang zu bent Funktionen über F2 auf. Schließlich untersuchen wir perfekte Sequenzen über Fp. Wir beweisen, dass diese äquivalent zu verallgemeinerten Differenzmengen sind. Es stellt sich her- aus, dass die klassischen Beispiele Projektionen verschiedener Typen verallge- meinerter Differenzmengen sind, die von desarguesschen Ebenen abgeleitet wer- den. Wir präsentieren auch einige Aussagen zur Nichtexistenz. i Abstract The topic of this dissertation are several types of generalized difference sets in abelian groups. First, we concentrate on (q, q, q, 1)-relative difference sets with q odd, which are equivalent to planar functions over Fq. Most of the known examples lead to commutative semifields. One of our main results is the construction of a new family of semifields of odd characteristic. We determine their left, right and middle nuclei. We show that for certain parameters, some of these semifields lead to two inequivalent planar functions. By applying the character approach to planar functions over Fq with q odd, we present some unexpected links between planar functions over Fpm and planar functions over Fp2m . We study the projections and liftings of planar functions, where we also apply a coding theory approach. Some computational results about the switchings of planar functions on F3n for n = 3, 4, 5, 6 are given. n n n n n Then we turn to (2 , 2 , 2 , 1)-relative difference sets in C4 relative to C2 . Two representations are introduced, one of which is the counterpart of the clas- sical planar functions defined on Fq with q odd. In more detail, a function n n n f on F2n leads to a (2 , 2 , 2 , 1)-relative difference set if and only if for any nonzero a, the mapping x 7! f (x + a) + f (x) + f (a) + xa is a permutation on F2n . We also call such f planar functions. We show that a planar func- tion f on F2n can be written as a Dembowski-Ostrom polynomial if and only if x ∗ a := f (x + a) + f (x) + f (a) + xa defines the multiplication of a semifield of or- n der 2 . When f is a mapping on F2n satisfying f (0) = 0 and Im( f ) = f0, xg with x 6= 0, we prove that f is a planar mapping if and only if f (x + y) = f (x) + f (y) for any x, y 2 F2n . We also consider projections of planar functions, which we call shifted-bent functions. They are shown to have an interesting relation with n bent functions over F2 . Finally, for any prime p, we investigate almost p-ary sequences which are perfect or nearly perfect. They are proved to be equivalent to certain types of generalized difference sets. The classical examples come from the projections of several types of generalized difference sets derived from desarguesian planes. We also present several nonexistence results. iii Acknowledgements First of all, I would like to express my gratitude to my supervisor Prof. Dr. Alexander Pott for the guidance and support. It is always enjoyable for me to discuss mathematics with him. His insight and patience helped me to discover many interesting results and to finish this dissertation. I am also grateful to Prof. Dr. Jürgen Bierbrauer, Dr. Ayça Çe¸smelio˘glo,Dr. Anton Malevich, Dr. Gohar Kyureghyan, Prof. Dr. Ferrüh Özbudak, Dr. Kai-Uwe Schmidt, Wei Su, Dr. Yin Tan, Carsten Thiel, Dr. Qi Wang and Prof. Dr. Wolfgang Willems for fruitful discussions, valuable comments and joyful collaborations. I am further grateful to other people who had indirect influence to my work: To my colleagues for their kindness friendship and support, together with their help on many occasions, to my former Master thesis supervisor Prof. Dr. Chao Li, who taught me finite fields. My deep gratitude goes to my parents and my brother Kai for all their love and encouragement. Finally, but not least, I would like to acknowledge the financial, academic and technical support from the Otto-von-Guericke University of Magdeburg and the financial support from CSC (China scholarship council). iv Contents Overview1 1 Projective planes and related algebraic structures5 1.1 Incidence structures............................ 5 1.2 Difference sets and their generalization ................ 8 1.3 Group rings and characters ....................... 11 1.4 Finite fields................................. 13 1.5 Projective and affine planes ....................... 16 1.6 Collineation and coordinatization.................... 19 1.7 Quasifields................................. 21 1.8 Semifields ................................. 24 1.9 Dembowski-Piper classification..................... 30 1.10 Projections and liftings of difference sets ............... 38 1.11 CCZ and EA equivalences........................ 41 2 A new family of semifields with two parameters 45 2.1 A family of commutative semifields .................. 45 2.2 Left and middle nuclei.......................... 48 2.3 The isotopism between Sk,s ....................... 54 2.4 Sk,s is a new family............................ 64 2.5 APN functions of a similar form .................... 66 3 Two approaches to planar functions 69 3.1 A character approach to planar functions............... 69 3.2 Switchings of planar functions ..................... 72 4 (q, q, q, 1)-relative difference sets with q even 85 4.1 Planar functions over F2n ........................ 86 4.2 Coordinatization.............................. 93 4.3 Nonexistence results for Boolean planar functions.......... 99 4.4 Shifted-bent functions ..........................103 4.5 Notes....................................111 5 Sequences derived from projective planes 115 5.1 Algebraic-combinatorial tools......................117 5.2 Almost p-ary perfect sequences.....................118 5.3 Almost p-ary nearly perfect sequences . 123 5.4 Appendix..................................126 Bibliography 133 Index 145 Notation 147 List of Tables 3.1 Weight distribution of Cf for odd n ................... 75 3.2 Weight distribution of Cf for even n . 76 3 2 3.3 Switching neighbors with respect to all l : F3 ! F3, F3 . 80 3.4 All known inequivalent planar functions over F34 . 81 4 3 2 3.5 Switching Neighbors with respect to all l : F3 ! F3, F3, F3 . 81 3.6 All known inequivalent planar functions over F35 . 81 5 4 3.7 Number of inequivalent l ◦ f for all l : F3 ! F3 . 82 5 3 3.8 Switching neighbors with respect to all l : F3 ! F3 . 82 5 2 3.9 Switching neighbors with respect to all l : F3 ! F3 . 82 3.10 All known inequivalent planar functions over F36 . 83 6 5 3.11 Number of inequivalent l ◦ f for all l : F3 ! F3 . 83 5.1 Orbits of G under x 7! x2 . 122 5.2 Existence Status of Perfect Sequences . 126 5.3 Existence Status of Nearly Perfect Sequences . 128 List of Figures 1.1 Fano plane................................. 17 4.1 Steps (iv) and (v) ............................. 94 Overview In the present thesis, we are interested in generalized difference sets, the devel- opments of which can be uniquely extended to projective planes. In Chapter1, in addition to fundamental definitions and results, I attempt to provide a comprehensive tour, from projective planes to generalized difference sets. After a brief introduction to the foundations of incidence structures, differ- ence sets and finite fields, in Section 1.5 we start to focus on projective planes, which are quite special incidence structures. A projective plane is an incidence structure with points and lines such that any two distinct points (resp.
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