Mustafa Gezek
In this study we performed a computer search for unitals in planes of order 16. Some new unitals were found and we show that some unitals can be embedded in two or more different planes.
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We show that the Ree unital $\mathcal{R}(q)$ has an embedding in a projective plane over a field $F$ if and only if $q=3$ and $\mathbb{F}_8$ is a subfield of $F$. In this case, the embedding is unique up to projective linear transformations. Besides elementary calculations, our proof uses the classification of the maximal subgroups of the simple Ree groups.
In this article we present the largest set of unitals (totally 553) in projective planes of order 16. An open question is what is the number of the known unitals that are non-isomorphic to the reported ones. The results are obtained with a program that implements author's algorithm.
In this paper we present the results from a program developed by the author that finds the unitals of the known 193 projective planes of order 25.. There are several planes for which we have not found any unital. One or more than one unitals have been found for most of the planes. The found unitals for a given plane are nonisomorphic each other. There are a few unitals isomorphic to a unital of another plane. A t - (v; k; λ) design D is a set X of points together with a family B of k-subsets of X called blocks with the property that every t points are contained in exactly λ blocks. The design …
Inspired by the connection between ovoids and unitals arising from the Buekenhout construction in the André/Bruck-Bose representation of translation planes of dimension at most two over their kernel, and since eggs of PG(4m-1,q), m>=1, are a generalization of ovoids, we explore the relation between eggs and unitals in translation planes of higher dimension over their kernel. By investigating such a relationship, we construct a unital in the Dickson semifield plane of order 3^{10}, which is represented in PG(20,3) by a cone whose base is a set of points constructed from the dual of the Penttila…
A finite shift plane can be equivalently defined via abelian relative difference sets as well as planar functions. In this paper, we present a generic way to construct unitals in finite shift planes of odd orders $q^2$. We investigate various geometric and combinatorial properties of them, such as the self-duality, the existences of O'Nan configurations, the Wilbrink's conditions, the designs formed by circles and so on. We also show that our unitals are inequivalent to the unitals derived from unitary polarities in the same shift planes. As designs, our unitals are also not isomorphic to the …
A Note on Fano Planes in Orthogonal Buekenhout-Metz Unitals of Even Order
该研究拓展了Feng和Li关于O'Nan构型的工作，将相关构造推广到偶阶正交Buekenhout-Metz酉平面的Fano平面中，发现了新的O'Nan构型并提出相关猜想。
A classification of unitals in nearfield planes with maximal automorphism group
本文对近场平面中具有最大自 automorphism 组的抛物单元式进行了分类，并探讨了更一般的结果。
Let $U_θ$ be a unital defined in a shift plane of odd order $q^2$, which are constructed recently by the authors. In particular, when the shift plane is desarguesian, $U_θ$ is a special Buekenhout-Metz unital formed by a union of ovals. We investigate the dimensions of the binary codes derived from $U_θ$. By using Kloosterman sums, we obtain a new lower bound on the aforementioned dimensions which improves the result obtained by Leung and Xiang in 2009. In particular, for $q=3^m$, this new lower bound equals $\frac{2}{3}(q^3+q^2-2q)-1$ for even $m$ and $\frac{2}{3}(q^3+q^2+q)-1$ for odd $m$.
Bishnoi conjectured that if a minimal t-fold blocking set in a projective plane of prime power order has maximal size then it is either a projective plane minus one point, the complement of a Baer subplane or a unital. In this note we prove this conjecture.
Let $\cU$ be a unital embedded in the Desarguesian projective plane $\PG(2,q^2)$. Write $M$ for the subgroup of $\PGL(3,q^2)$ which preserves $\cU$. We show that $\cU$ is classical if and only if $\cU$ has two distinct points $P,Q$ for which the stabiliser $G=M_{P,Q}$ has order $q^2-1$.
This article studies a generalization of magic squares to finite projective planes. In traditional magic squares the entries come from the natural numbers. This does not work for finite projective planes, so we instead use Abelian groups. For each finite projective plane we demonstrate a small group over which the plane can labeled magically. In the prime order case we classify all groups over which the projective plane can be made magic.
This paper studies the existence of finite non-Desarguesian flag-transitive projective plane, giving necessary conditions in terms of polynomial equations over finite fields of characteristic $3$. This sheds light on the longstanding conjecture that every finite flag-transitive projective plane is Desarguesian.
Projective Hjelmslev planes and Affine Hjelmselv planes are generalisations of projective planes and affine planes. We present an algorithm for constructing a projective Hjelmslev planes and affine Hjelsmelv planes using projective planes, affine planes and orthogonal arrays. We show that all 2-uniform projective Hjelmslev planes, and all 2-uniform affine Hjelsmelv planes can be constructed in this way. As a corollary it is shown that all 2-uniform Affine Hjelmselv planes are sub-geometries of 2-uniform projective Hjelmselv planes.
We take points and planes as fundamental, lines as derived, in an axiomatic formulation of three-dimensional projective space, the self-dual nature of which formulation renders automatic the principle of duality.
We prove that a triangulation of the projective plane is (strongly) $t$-perfect if and only if it is perfect and contains no $K_4$.
The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising frommaximal (120,8)-arcs in the known projective planes of order 16 are computed. It is shown that each of these designs is embeddable in a unique way in a projective plane of order 16.
We study combinatorial parameters of a recently introduced bootstrap percolation problem in finite projective planes. We present sharp results on the size of the minimum percolating sets and the maximal non-percolating sets. Additional results on the minimal and maximal percolation time as well as on the critical probability in the projective plane are also presented.
We build unbounded classes of plane and projective plane multiwheels that are 4-critical that are received summing odd wheels as edge sums modulo two. These classes can be considered as ascending from single common graph that can be received as edge sum modulo two of the octahedron graph O and the minimal wheel W3. All graphs of these classes belong to 2n-2-edges-class of graphs, among which are those that quadrangulate projective plane, i.e., graphs from Grötzsch class, received applying Mycielski's Construction to odd cycle.
We apply an improvement of the Delsarte LP-bound to give a new proof of the non-existence of finite projective planes of order 6, and uniqueness of finite projective planes of order 7. The proof is computer aided, and it is also feasible to apply to higher orders like 8, 9 and, with further improvements, possibly 10 and 12.
We construct a new family of minimal non-orientable matroids of rank three. Some of these matroids embed in Desarguesian projective planes. This answers a question of Ziegler: for every prime power $q$, find a minimal non-orientable submatroid of the projective plane over the $q$-element field.
Let $n$ be the order of a (quaternary) Hadamard matrix. It is shown that the existence of a projective plane of order $n$ is equivalent to the existence of a balancedly multi-splittable (quaternary) Hadamard matrix of order $n^2$.
In this paper, we prove that the crossing number of circulant graph $C(3k+1;\{1,k\})$ on the projective plane is $k$ for $k \geq 3$.
We introduce and study embeddings of graphs in finite projective planes, and present related results for some families of graphs including complete graphs and complete bipartite graphs. We also make connections between embeddings of graphs and the existence of certain substructures in a plane, such as Baer subplanes and arcs.
We determine the partition dimension of the incidence graph $G(Π_q)$ of the projective plane $Π_q$ up to a constant factor $2$ as $(2+o(1))\log_2{q}\leq \mathrm{pd}(G(Π_q))\leq (4+o(1))\log_2{q}.$
Spanning bipartite quadrangulations of triangulations of the projective plane
本文研究了射影平面三角形的生成双色四边形化子图特性，解决了Kündgen和Ramamurthi在2002年提出的问题。
A combinatorial characterization of resolvable Steiner 2-$(v,k,1)$ designs embeddable as maximal arcs in a projective plane of order $(v-k)/(k-1)$ is proved, and a generalization of a conjecture by Andries Brouwer \cite{Br} is formulated.
The minimum weight of the code generated by the incidence matrix of points versus lines in a projective plane has been known for over 50 years. Surprisingly, finding the minimum weight of the dual code of projective planes of non-prime order is still an open problem, even in the Desarguesian case. In this paper, we focus on the case of projective planes of order $p^2$, where $p$ is prime, and we link the existence of small weight code words in the dual code to the existence of embedded subplanes and {\em antipodal planes}. In the Desarguesian case, we can exclude such code words by showing a m…
We prove that the entropy compression method systematized by L. Esperet and A. Parreau [11] can be applied to any problem formulated in the variable version of the Lovász Local Lemma. As an application, we prove the existence of legitimate colorings for projective planes with small orders, which extends results of N. Alon and Z. Füredi [2]. In fact, we allow different numbers of colors, proving that projective planes of any order can be legitimate colored with 42 colors.
An $n$-arc in a projective plane is a collection of $n$ distinct points in the plane, no three of which lie on a line. Formulas counting the number of $n$-arcs in any finite projective plane of order $q$ are known for $n \le 8$. In 1995, Iampolskaia, Skorobogatov, and Sorokin counted $9$-arcs in the projective plane over a finite field of order $q$ and showed that this count is a quasipolynomial function of $q$. We present a formula for the number of $9$-arcs in any projective plane of order $q$, even those that are non-Desarguesian, deriving Iampolskaia, Skorobogatov, and Sorokin's formula as…
In this paper we investigate light dual multinets labeled by a finite group in the projective plane $PG(2,\mathbb{K})$ defined over a field $\mathbb{K}$. We present two classes of new examples. Moreover, under some conditions on the characteristic $\mathbb{K}$, we classify group-labeled light dual multinets with lines of length least $9$.
This report gives an overview of the history of finite projective planes and their properties before going on to outline the proof that no projective plane of order 10 exists. The report also investigates the search carried out by MacWilliams, Sloane and Thompson in 1970 [12] and confirms their result by providing independent verification that there is no vector of weight 15 in the code generated by the projective plane of order 10.
We study Poncelet's Theorem in the four non-isomorphic finite projective planes of order 9. Among these planes, only the Desarguesian plane turns out to be a Poncelet plane, while the other three planes which are constructed over the miniquaternion near-field of order 9, are not. This gives a complete discussion of Poncelet's Theorem in finite projective planes of order 9.
I adapt Berlekamp's light bulb switching game to finite projective plans and finite affine planes, then find the worst arrangement of lit bulbs for planes of even and odd orders. The results are then extended from the planes to spaces of higher dimension.
Toroidal embeddings of cubic projective plane obstructions
本文对六个三次射影平面障碍图在环面中的嵌入进行了等价分类，完善了 Glover 和 Huneke 关于三次图射影平面嵌入障碍的结果。
A long-standing conjecture is that any transitive finite projective plane is Desarguesian. We make a contribution towards a proof of this conjecture by showing that a group acting transitively on the the points of a non-Desarguesianprojective plane must not contain any components.
We geometrically characterise the Veronese representations of ring projective planes over algebras which are analogues of the dual numbers, giving rise to projective Hjelmslev planes of level 2 coordinatised over quadratic alternative algebras. These planes are related to affine buildings of relative type $\widetilde{A}_2$ and respective absolute type $\widetilde{\mathsf{A}}_2$, $\widetilde{\mathsf{A}}_5$ and $\widetilde{\mathsf{E}}_6$.
A pair of planes, both projective or both affine, of the same order and on the same pointset are orthogoval if each line of one plane intersects each line of the other plane in at most two points. In this paper we prove new constructions for sets of mutually orthogoval planes, both projective and affine, and review known results that are equivalent to sets of more than two mutually orthogoval planes. We also discuss the connection between sets of mutually orthogoval planes and covering arrays.
The aim of this paper is twofold: First we classify all abstract light dual multinets of order $6$ which have a unique line of length at least two. Then we classify the weak projective embeddings of these objects in projective planes over fields of characteristic zero. For the latter we present a computational algebraic method for the study of weak projective embeddings of finite point-line incidence structures.
Suppose that a group $G$ acts transitively on the points of $\mathcal{P}$, a finite non-Desarguesian projective plane. We prove that if $G$ is insoluble then $G/O(G)$ is isomorphic to $SL_2(5)$ or $SL_2(5).2$.
Welschinger invariants of the real projective plane can be computed via the enumeration of enriched graphs, called marked floor diagrams. By a purely combinatorial study of these objects, we prove a Caporaso-Harris type formula which allows one to compute Welschinger invariants for configurations of points with any number of complex conjugated points.
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