14 research outputs found

    On Buekenhout-Metz unitals of even order

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    AbstractThe even order Buekenhout-Metz unitals are enumerated (up to projective equivalence) and their inherited collineation groups are computed. They are shown to be self-dual as designs, and certain related designs are also constructed

    An alternative construction of B-M and B-T unitals in Desarguesian planes

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    We present a new construction of non-classical unitals from a classical unital UU in PG(2,q2)PG(2,q^2). The resulting non-classical unitals are B-M unitals. The idea is to find a non-standard model Π\Pi of PG(2,q2)PG(2,q^2) with the following three properties: 1. points of Π\Pi are those of PG(2,q2)PG(2,q^2); 2. lines of Π\Pi are certain lines and conics of PG(2,q2)PG(2,q^2); 3. the points in UU form a non-classical B-M unital in Π\Pi. Our construction also works for the B-T unital, provided that conics are replaced by certain algebraic curves of higher degree.Comment: Keywords: unital, desarguesian plane 11 pages; ISSN: 0012-365

    On the Equivalence, Stabilisers, and Feet of Buekenhout-Tits Unitals

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    This paper addresses a number of problems concerning Buekenhout-Tits unitals in PG(2,q2)PG(2,q^2), where q=2e+1q = 2^{e+1} and e≄1e \geq 1. We show that all Buekenhout-Tits unitals are PGLPGL-equivalent (addressing an open problem in [S. Barwick and G. L. Ebert. Unitals in projective planes. Springer Monographs in Mathematics. Springer, New York, 2008.]), explicitly describe their PΓLP\Gamma L-stabiliser (expanding Ebert's work in [G.L. Ebert. Buekenhout-Tits unitals. J. Algebraic. Combin. 6.2 (1997), 133-140], and show that lines meet the feet of points no on ℓ∞\ell_\infty in at most four points. Finally, we show that feet of points not on ℓ∞\ell_\infty are not always a {0,1,2,4}\{0,1,2,4\}-set, in contrast to what happens for Buekenhout-Metz unitals [N. Abarz\'ua, R. Pomareda, and O. Vega. Feet in orthogonal-Buekenhout-Metz unitals. Adv. Geom. 18.2 (2018), 229-236]

    Embedding of orthogonal Buekenhout-Metz unitals in the Desarguesian plane of order q^2

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    A unital, that is a 2-(q^3 + 1, q + 1, 1) block-design, is embedded in a projective plane π of order q^2 if its points are points of π and its blocks are subsets of lines of π, the point-block incidences being the same as in π. Regarding unitals U which are isomorphic, as a block-design, to the classical unital, T. Szonyi and the authors recently proved that the natural embedding is the unique embedding of U into the Desarguesian plane of order q^2. In this paper we extend this uniqueness result to all unitals which are isomorphic, as block-designs, to orthogonal Buekenhout-Metz unitals

    On regular sets of affine type in finite Desarguesian planes and related codes

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    In this paper, we consider point sets of finite Desarguesian planes whose multisets of intersection numbers with lines is the same for all but one exceptional parallel class of lines. We call such sets regular of affine type. When the lines of the exceptional parallel class have the same intersection numbers, then we call these sets regular of pointed type. Classical examples are e.g. unitals; a detailed study and constructions of such sets with few intersection numbers is due to Hirschfeld and Sz\H{o}nyi from 1991. We here provide some general construction methods for regular sets and describe a few infinite families. The members of one of these families have the size of a unital and meet affine lines of PG(2,q2)\mathrm{PG}(2, q^2) in one of 44 possible intersection numbers, each of them congruent to 11 modulo q\sqrt{q}. As a byproduct, we determine the intersection sizes of the Hermitian curve defined over GF(q2)\mathrm{GF}(q^2) with suitable rational curves of degree q\sqrt{q} and we obtain q\sqrt{q}-divisible codes with 55 non-zero weights. We also determine the weight enumerator of the codes arising from the general constructions modulus some qq-powers.Comment: 16 pages/revised and improved versio
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