188 research outputs found

    Characterisations of elementary pseudo-caps and good eggs

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    In this note, we use the theory of Desarguesian spreads to investigate good eggs. Thas showed that an egg in PG(4n1,q)\mathrm{PG}(4n-1, q), qq odd, with two good elements is elementary. By a short combinatorial argument, we show that a similar statement holds for large pseudo-caps, in odd and even characteristic. As a corollary, this improves and extends the result of Thas, Thas and Van Maldeghem (2006) where one needs at least 4 good elements of an egg in even characteristic to obtain the same conclusion. We rephrase this corollary to obtain a characterisation of the generalised quadrangle T3(O)T_3(\mathcal{O}) of Tits. Lavrauw (2005) characterises elementary eggs in odd characteristic as those good eggs containing a space that contains at least 5 elements of the egg, but not the good element. We provide an adaptation of this characterisation for weak eggs in odd and even characteristic. As a corollary, we obtain a direct geometric proof for the theorem of Lavrauw

    Pseudo-ovals in even characteristic and ovoidal Laguerre planes

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    Pseudo-arcs are the higher dimensional analogues of arcs in a projective plane: a pseudo-arc is a set A\mathcal{A} of (n1)(n-1)-spaces in PG(3n1,q)\mathrm{PG}(3n-1,q) such that any three span the whole space. Pseudo-arcs of size qn+1q^n+1 are called pseudo-ovals, while pseudo-arcs of size qn+2q^n+2 are called pseudo-hyperovals. A pseudo-arc is called elementary if it arises from applying field reduction to an arc in PG(2,qn)\mathrm{PG}(2,q^n). We explain the connection between dual pseudo-ovals and elation Laguerre planes and show that an elation Laguerre plane is ovoidal if and only if it arises from an elementary dual pseudo-oval. The main theorem of this paper shows that a pseudo-(hyper)oval in PG(3n1,q)\mathrm{PG}(3n-1,q), where qq is even and nn is prime, such that every element induces a Desarguesian spread, is elementary. As a corollary, we give a characterisation of certain ovoidal Laguerre planes in terms of the derived affine planes

    On Hyperfocused Arcs in PG(2,q)

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    A k-arc in a Dearguesian projective plane whose secants meet some external line in k-1 points is said to be hyperfocused. Hyperfocused arcs are investigated in connection with a secret sharing scheme based on geometry due to Simmons. In this paper it is shown that point orbits under suitable groups of elations are hyperfocused arcs with the significant property of being contained neither in a hyperoval, nor in a proper subplane. Also, the concept of generalized hyperfocused arc, i.e. an arc whose secants admit a blocking set of minimum size, is introduced: a construction method is provided, together with the classification for size up to 10

    On Mathon's construction of maximal arcs in Desarguesian planes. II

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    In a recent paper [M], Mathon gives a new construction of maximal arcs which generalizes the construction of Denniston. In relation to this construction, Mathon asks the question of determining the largest degree of a non-Denniston maximal arc arising from his new construction. In this paper, we give a nearly complete answer to this problem. Specifically, we prove that when m5m\geq 5 and m9m\neq 9, the largest dd of a non-Denniston maximal arc of degree 2d2^d in PG(2,2^m) generated by a {p,1}-map is (\floor {m/2} +1). This confirms our conjecture in [FLX]. For {p,q}-maps, we prove that if m7m\geq 7 and m9m\neq 9, then the largest dd of a non-Denniston maximal arc of degree 2d2^d in PG(2,2^m) generated by a {p,q}-map is either \floor {m/2} +1 or \floor{m/2} +2.Comment: 21 page
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