1,706 research outputs found
Abstract hyperovals, partial geometries, and transitive hyperovals
Includes bibliographical references.2015 Summer.A hyperoval is a (q+2)- arc of a projective plane π, of order q with q even. Let G denote the collineation group of π containing a hyperoval Ω. We say that Ω is transitive if for any pair of points x, y is an element of Ω, there exists a g is an element of G fixing Ω setwise such that xg = y. In1987, Billotti and Korchmaros proved that if 4||G|, then either Ω is the regular hyperoval in PG(2,q) for q=2 or 4 or q = 16 and |G||144. In 2005, Sonnino proved that if |G| = 144, then π is desarguesian and Ω is isomorphic to the Lunelli-Sce hyperoval. For our main result, we show that if G is the collineation group of a projective plane containing a transitivehyperoval with 4 ||G|, then |G| = 144 and Ω is isomorphic to the Lunelli-Sce hyperoval. We also show that if A(X) is an abstract hyperoval of order n ≡ 2(mod 4); then |Aut(A(X))| is odd. If A(X) is an abstract hyperoval of order n such that Aut(A(X)) contains two distinct involutions with |FixX(g)| and |FixX(ƒ)| ≥ 4. Then we show that FixX(g) ≠FixX(ƒ). We also show that there is no hyperoval of order 12 admitting a group whose order is divisible by 11 or 13, by showing that there is no partial geometry pg(6, 10, 5) admitting a group of order 11 or of order 13. Finally, we were able to show that there is no hyperoval in a projective plane of order 12 with a dihedral subgroup of order 14, by showing that that there is no partial geometry pg(7, 12, 6) admitting a dihedral group of order 14. The latter results are achieved by studying abstract hyperovals and their symmetries
Linear sections of GL(4, 2)
For V = V (n; q); a linear section of GL(V ) = GL(n; q) is a vector subspace S of the n 2 -dimensional vector space End(V ) which is contained in GL(V ) [ f0g: We pose the problem, for given (n; q); of classifying the di erent kinds of maximal linear sections of GL(n; q): If S is any linear section of GL(n; q) then dim S n: The case of GL(4; 2) is examined fully. Up to a suitable notion of equiv- alence there are just two classes of 3-dimensional maximal normalized linear sections M3;M0 3 , and three classes M4;M0 4 ;M00 4 of 4-dimensional sections. The subgroups of GL(4; 2) generated by representatives of these ve classes are respectively G3 = A7; G 0 3 = GL(4; 2); G4 = Z15; G 0 4 = Z3 A5; G 00 4 = GL(4; 2): On various occasions use is made of an isomorphism T : A8 ! GL(4; 2): In particular a representative of the class M3 is the image under T of a subset f1; ::: ; 7g of A7 with the property that 1 i j is of order 6 for all i =6 j: The classes M3;M0 3 give rise to two classes of maximal partial spreads of order 9 in PG(7; 2); and the classes M0 4 ;M00 4 yield the two isomorphism classes of proper semi eld planes of order 16
Classification of large partial plane spreads in and related combinatorial objects
In this article, the partial plane spreads in of maximum possible
size and of size are classified. Based on this result, we obtain the
classification of the following closely related combinatorial objects: Vector
space partitions of of type , binary MRD
codes of minimum rank distance , and subspace codes with parameters
and .Comment: 31 pages, 9 table
Higgledy-piggledy sets in projective spaces of small dimension
This work focuses on higgledy-piggledy sets of -subspaces in
, i.e. sets of projective subspaces that are 'well-spread-out'.
More precisely, the set of intersection points of these -subspaces with any
-subspace of spans itself. We
highlight three methods to construct small higgledy-piggledy sets of
-subspaces and discuss, for , 'optimal' sets that cover the
smallest possible number of points. Furthermore, we investigate small
non-trivial higgledy-piggledy sets in , . Our main
result is the existence of six lines of in higgledy-piggledy
arrangement, two of which intersect. Exploiting the construction methods
mentioned above, we also show the existence of six planes of
in higgledy-piggledy arrangement, two of which maximally intersect, as well as
the existence of two higgledy-piggledy sets in consisting of
eight planes and seven solids, respectively. Finally, we translate these
geometrical results to a coding- and graph-theoretical context.Comment: [v1] 21 pages, 1 figure [v2] 21 pages, 1 figure: corrected minor
details, updated bibliograph
Constant rank-distance sets of hermitian matrices and partial spreads in hermitian polar spaces
In this paper we investigate partial spreads of through the
related notion of partial spread sets of hermitian matrices, and the more
general notion of constant rank-distance sets. We prove a tight upper bound on
the maximum size of a linear constant rank-distance set of hermitian matrices
over finite fields, and as a consequence prove the maximality of extensions of
symplectic semifield spreads as partial spreads of . We prove
upper bounds for constant rank-distance sets for even rank, construct large
examples of these, and construct maximal partial spreads of for a
range of sizes
Some results on spreads and ovoids
We survey some results on ovoids and spreads of finite polar spaces, focusing on the ovoids of H(3,q^2) arising from spreads of PG(3,q)
via indicator sets and Shult embedding, and on some related constructions.
We conclude with a remark on symplectic spreads of PG(2n-1,q)
The known maximal partial ovoids of size of Q(4,q)
We present a description of maximal partial ovoids of size of the
parabolic quadric \q(4,q) as sharply transitive subsets of \SL(2,q) and
show their connection with spread sets. This representation leads to an elegant
explicit description of all known examples. We also give an alternative
representation of these examples which is related to root systems.Comment: 15 pages, revised version (v2 on arxiv is just a update of another
paper, applied on this paper, clearly a mistake...
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