Intersections Of Hyperconics And Configurations In Classical Planes

Let {dollar}\pi{dollar} = PG(2,F), where F is a field of characteristic 2 and of order greater than 2. Given a conic, its tangents all pass through a common point, the nucleus. A conic, together with its nucleus, is called a hyperconic. All conics considered are non-degenerate.;First, a relationship is established between hyperconics and certain symmetric unipotent Latin squares for all finite projective planes.;Intersection properties of hyperconics in PG(2,F), Fano configurations containing points of a hyperconic, as well as certain subplanes of PG(2,F) are studied. An open question in {dollar}\pi{dollar} = PG(2,q), q even, is: what is the size and structure of a set or maximum size of hyperovals (or hyperconics) pairwise intersecting in exactly 2 points? In PG(2,4), such a set is shown to have size 16 and to have one of 2 \u27dual\u27 structures: 16 hyperconics missing a fixed line, or 16 hyperconics through a fixed point.;The former is a 2 {dollar}-{dollar} (16,6,2)-design of grid type which can be obtained from the 5 {dollar}-{dollar}(24,8,1) Mathieu design, and which can be related to singular points of a Kummer surface in PG(2,q) for q odd (see (Bruen 2)).;The latter is shown to be an affine plane in 2 ways: (i) taking the hyperconics which all contain the fixed point, as well as the lines through that fixed point (in the original plane) to be the lines of an AG(2,4); and (ii) taking the hyperconics in the original plane to be the points, and the points (except the fixed point in all 16 hyperconics) in the original plane to be the lines of an AG(2,4).;In PG(2,F) let the field F contain a subfield of order 4. Then, in PG(2,F) we describe certain sets of 6 points no 3 collinear called hexagons. It is then shown how the much studied even intersection property in PG(2,4) can be lifted (extended) to certain sets of hyperconics in PG(2,F)

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

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 $m\geq 5$ and $m\neq 9$, the largest $d$ of a non-Denniston maximal arc of degree $2^d$ 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 $m\geq 7$ and $m\neq 9$, then the largest $d$ of a non-Denniston maximal arc of degree $2^d$ 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

On symplectic semifield spreads of PG(5,q2), q odd

We prove that there exist exactly three non-equivalent symplectic semifield spreads of PG ( 5 , q2), for q2> 2 .38odd, whose associated semifield has center containing Fq. Equivalently, we classify, up to isotopy, commutative semifields of order q6, for q2> 2 .38odd, with middle nucleus containing q2Fq2and center containing q Fq

Optimal Binary Subspace Codes of Length 6, Constant Dimension 3 and Minimum Distance 4

It is shown that the maximum size of a binary subspace code of packet length $v=6$, minimum subspace distance $d=4$, and constant dimension $k=3$ is $M=77$; in Finite Geometry terms, the maximum number of planes in $\operatorname{PG}(5,2)$ mutually intersecting in at most a point is $77$. Optimal binary $(v,M,d;k)=(6,77,4;3)$ subspace codes are classified into $5$ isomorphism types, and a computer-free construction of one isomorphism type is provided. The construction uses both geometry and finite fields theory and generalizes to any $q$, yielding a new family of $q$-ary $(6,q^6+2q^2+2q+1,4;3)$ subspace codes

Combinatorial Intricacies of Labeled Fano Planes

Given a seven-element set $X = \{1,2,3,4,5,6,7\}$, there are 30 ways to define a Fano plane on it. Let us call a line of such Fano plane, that is to say an unordered triple from $X$, ordinary or defective according as the sum of two smaller integers from the triple is or is not equal to the remaining one, respectively. A point of the labeled Fano plane is said to be of order $s$, $0 \leq s \leq 3$, if there are $s$ {\it defective} lines passing through it. With such structural refinement in mind, the 30 Fano planes are shown to fall into eight distinct types. Out of the total of 35 lines, nine ordinary lines are of five different kinds, whereas the remaining 26 defective lines yield as many as ten distinct types. It is shown, in particular, that no labeled Fano plane can have all points of zeroth order, or feature just one point of order two. A connection with prominent configurations in Steiner triple systems is also pointed out.Comment: 5 pages, 2 figure

Classification of book spreads in PG(5, 2)

We classify all line spreads $\mathcal{S}_{21}$ in $\operatorname*{PG}(5,2)$ of a special kind, namely those which are \emph{book spreads}. We show that up to isomorphism there are precisely nine different kinds of book spreads and describe the automorphism groups which stabilize them. Most of the main results are obtained in two independent ways, namely theoretically and by computer. \en