Yazar "Gezek, Mustafa" seçeneğine göre listele

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    Disjoint maximal arcs in projective planes of order 16
    (Yildiz Technical Univ, 2024) Gezek, Mustafa
    This paper provides the results of some computer searches for disjoint maximal (52, 4) -arcs in the known planes of order 16. Thirty-seven new such sets are discovered: four in Johnson plane and thirty-three in Mathon plane, eighteen of which give examples of 104 -sets of type (4,8) coming from non -isomorphic pairs of maximal (52, 4) -arcs, providing first examples for such sets. A new lower bound on the number of 104 -sets of type (4,8) coming from disjoint maximal (52, 4) -arcs in the known planes of order 16 is obtained. The 104 -length binary and ternary linear codes generated by the blocks of 1 -designs associated with the known 104 -sets of type (4,8) are classified.
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    DISJOINT SETS IN PROJECTIVE PLANES OF SMALL ORDER
    (2023) Gezek, Mustafa
    In this paper, results of a computer search for disjoint sets associated with maximal arcs and unitals in projective planes of order 16, and disjoint sets associated with unitals in projective planes of orders 9 and 25 are reported. It is shown that the number of pairs of disjoint unitals in planes of order 9 is exactly four, and new pairs and triples of disjoint degree 4 maximal arcs are shown to exist in some of the planes of order 16. New bounds on the number of 104-sets of type (4, 8) and 156-sets of type (8, 12) are achieved. A combinatorial method for finding new maximal arcs, new unitals, and new v-sets of type (m, n) is introduced. All disjoint sets found in this study are explicitly listed.
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    DISJOINT SETS IN PROJECTIVE PLANES OF SMALL ORDER
    (Ankara Univ, Fac Sci, 2023) Gezek, Mustafa
    In this paper, results of a computer search for disjoint sets as-sociated with maximal arcs and unitals in projective planes of order 16, and disjoint sets associated with unitals in projective planes of orders 9 and 25 are reported. It is shown that the number of pairs of disjoint unitals in planes of order 9 is exactly four, and new pairs and triples of disjoint degree 4 maximal arcs are shown to exist in some of the planes of order 16. New bounds on the number of 104-sets of type (4,8) and 156-sets of type (8,12) are achieved. A combinatorial method for finding new maximal arcs, new unitals, and new v-sets of type (m, n) is introduced. All disjoint sets found in this study are explicitly listed.
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    Maximal arcs in projective planes of order 16 and related designs
    (Walter De Gruyter Gmbh, 2019) Gezek, Mustafa; Tonchev, Vladimir D.; Wagner, Tim
    The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising from maximal (120,8)-arcs, and the 2-(52, 4, 1) designs arising from previously known maximal (52, 4)-arcs, as well as some newly discovered maximal (52, 4)-arcs in the known projective planes of order 16, are computed. It is shown that each 2-(1.20,8,1) design associated with a maximal (1.20,8)-arc is embeddable in a unique way in a projective plane of order 16. This result suggests a possible strengthening of the Bose-Shrikhande theorem about the embeddability of the complement of a hyperoval in a projective plane of even order. The computations of the maximal sets of compatible resolutions of the 2-(52, 4, 1) designs associated with maximal (52,4)-arcs show that five of the known projective planes of order 16 contain maximal arcs whose associated designs are embeddable in two nonisomorphic planes of order 16.
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    Maximal arcs, codes, and new links between projective planes of order 16
    (Electronic Journal Of Combinatorics, 2020) Gezek, Mustafa; Mathon, Rudi; Tonchev, Vladimir D.
    In this paper we consider binary linear codes spanned by incidence matrices of Steiner 2-designs associated with maximal arcs in projective planes of even order, and their dual codes. Upper and lower bounds on the 2-rank of the incidence matrices are derived. A lower bound on the minimum distance of the dual codes is proved, and it is shown that the bound is achieved if and only if the related maximal arc contains a hyperoval of the plane. The binary linear codes of length 52 spanned by the incidence matrices of 2-(52, 4, 1) designs associated with previously known and some newly found maximal arcs of degree 4 in projective planes of order 16 are analyzed and classified up to equivalence. The classification shows that some designs associated with maximal arcs in nonisomorphic planes generate equivalent codes. This phenomenon establishes new links between several of the known planes. A conjecture concerning the codes of maximal arcs in PG(2, 2(m)) is formulated.
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    On partial geometries arising from maximal arcs
    (Springer, 2022) Gezek, Mustafa; Tonchev, Vladimir D.
    The subject of this paper are partial geometries pg(s, t, ?) with parameters s=d(d?-1),t=d?(d-1),?=(d-1)(d?-1), d, d?? 2. In all known examples, q= dd? is a power of 2 and the partial geometry arises from a maximal arc of degree d or d? in a projective plane of order q via a known construction due to Thas [28] and Wallis [34], with a single known exception of a partial geometry pg(4, 6, 3) found by Mathon [22] that is not associated with a maximal arc in the projective plane of order 8. A parallel class of lines is a set of pairwise disjoint lines that covers the point set. Two parallel classes are called orthogonal if they share exactly one line. An upper bound on the maximum number of pairwise orthogonal parallel classes in a partial geometry G with parameters pg(d(d?- 1) , d?(d- 1) , (d- 1) (d?- 1)) is proved, and it is shown that a necessary and sufficient condition for G to arise from a maximal arc of degree d or d? in a projective plane of order q= dd? is that both G and its dual geometry contain sets of pairwise orthogonal parallel classes that meet the upper bound. An alternative construction of Mathon’s partial geometry is presented, and the new necessary condition is used to demonstrate why this partial geometry is not associated with any maximal arc in the projective plane of order 8. The partial geometries associated with all known maximal arcs in projective planes of order 16 are classified up to isomorphism, and their parallel classes of lines and the 2-rank of their incidence matrices are computed. Based on these results, some open problems and conjectures are formulated. © 2021, Springer Science+Business Media, LLC, part of Springer Nature.
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    Pedal Sets of Unitals in Projective Planes of Order 16
    (2022) Gezek, Mustafa
    In this article, we perform computer searches for pedal sets of all known unitals in the known planes of order 16. Special points of unitals having at least one special tangent are studied in detail. It is shown that unitals without special points exist. Open problems regarding the computational results presented in this study are discussed. A conjecture about the numbers of line types of an unital $U$ and its dual unital $U^perp$ is formulated.
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    Secant Distributions of Unitals
    (Springer Basel Ag, 2024) Gezek, Mustafa
    Let U be a unital embedded in a projective plane Pi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi $$\end{document} of order q2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q<^>2$$\end{document}. For R is an element of U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R\in U$$\end{document}, let sR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_R$$\end{document} and tR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_R$$\end{document} be a secant line through R and the tangent line to U at point R, respectively. If the tangent lines to U, passing through the points in sR boolean AND U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_R\cap U$$\end{document}, intersect at a single point on tR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_R$$\end{document}, then sR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_R$$\end{document} is referred to as a secant line satisfying the desired property. If ni\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_i$$\end{document} of the points of U have exactly mi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_i$$\end{document} secant lines satisfying the desired property, then m1n1,m2n2,& ctdot;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} m_1<^>{n_1}, m_2<^>{n_2}, \cdots \end{aligned}$$\end{document}is called the secant distribution of U, where & sum;ni=q3+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum n_i=q<^>3+1$$\end{document}, and 0 <= mi <= q2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le m_i\le q<^>2$$\end{document}. In this article, we show that collinear pedal sets of a unital U plays an important role in the secant distribution of U. Formulas for secant distributions of unitals having 0,1,q2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0,1,q<^>2,$$\end{document} or q2+q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q<^>2+q$$\end{document} special points are provided. Statistics regarding to secant distributions of unitals embedded in planes of orders q2 <= 25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q<^>2\le 25$$\end{document} are presented. Some open problems related to secant distributions of unitals having specific number of collinear pedal sets are discussed.
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    Unitals in projective planes of order 16
    (Scientific Technical Research Council Turkey-Tubitak, 2021) Stoichev, Stoicho Dimitrov; Gezek, Mustafa
    In this study, we perform computer searches for unitals in planes of order 16. The number of known nonisomorphic unitals in these planes is improved to be 261. Some data related to 2- (65, 5, 1) designs associated with unitals are given. New lower bounds on the number of unital designs in projective planes of order 16 and 2- (65, 5, 1) designs are established. The computations show that thirty-nine unitals can be embedded in two or more nonisomorphic projective planes of order 16. Fifteen new connections between planes of order 16 (based on unitals) are found. All unitals found by the algorithms used in this study are explicitly listed. We assume familiarity with the basic facts from combinatorial design theory and finite geometries [5, 9, 16]. A t-(v, k, ?) design (t-design) is a pair D = {X, B} of a set X of cardinality v, called points, and a collection B of k-subsets of X, called blocks, such that every t points appear together in exactly ? blocks. A 2-design with ? = 1 is called a Steiner design. The incidence matrix of a 2-(v, k, ?) design D is a matrix M = (mij) with rows labeled by the blocks of D, columns labeled by the points of D, where mi,j = 1 if the ith block contains the j th point and mi,j = 0 otherwise. For a prime p, the rank of the incidence matrix of design D over a finite field of characteristic p is called the p-rank of D. Two designs D and D? are called isomorphic if there is a bijection between their point sets that maps
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    Unitals in Projective Planes of Order 25
    (Springer Basel Ag, 2023) Stoichev, Stoicho D. D.; Gezek, Mustafa
    In this paper, results of a non-exhaustive computer search for unitals in the known planes of order twenty-five are reported. The 2-(126, 6, 1) designs associated with newly found unitals are studied in detail. 938 non-isomorphic unital designs are discovered and we show that three of the unital designs are embeddable in two non-isomorphic planes and 239 of them are resolvable. The findings of this study improve some well-known lower bounds on the number of such designs and provide new connections between some pairs of planes. A conjecture concerning the p-ranks of unital designs embedded in planes of order q(2) is formulated.