Yazar "Tonchev, Vladimir D." seçeneğine göre listele

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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.