Ambient Spaces of Dimensional Dual Arcs

A d-dimensional dual arc in PG(n, q) is a higher dimensional analogue of a dual arc in a projective plane. For every prime power q other than 2, the existence of a d-dimensional dual arc (d 2) in PG(n, q) of a certain size implies n d(d + 3)/2 (Theorem 1). This is best possible, because of the recent construction of d-dimensional dual arcs in PG(d(d + 3)/2, q) of size ^d–1 _i=0 q ⁱ, using the Veronesean, observed first by Thas and van Maldeghem (Proposition 7). Another construction using caps is given as well (Proposition 10).     

On an Isomorphism Problem of Some Dual Hyperovals in PG(2d + 1, q) with q even

Let q = 2^l with l≥ 1 and d ≥ 2. We prove that any automorphism of the d-dimensional dual hyperoval over GF(q), constructed in [3] for any (d + 1)-dimensional GF(q)-vector subspace V in GF(qⁿ) with n≥ d + 1 and for any generator σ of the Galois group of GF(qⁿ) over GF(q), always fixes the special member X(∞). Moreover, we prove that, in case V = GF(q^d+1), two dual hyperovals and in PG(2d + 1,q), where σ and τ are generators of the Galois group of GF(q^d+1) over GF(q), are isomorphic if and only if (1) σ = τ or (2) σ τ = id. Therefore, we have proved that, even in the case q > 2, there exist non isomorphic d-dimensional dual hyperovals in PG(2d + 1,q) for d ≥ 3.     

On the Number of k‐Dominating Independent Sets

We study the existence and the number of k‐dominating independent sets in certain graph families. While the case namely the case of maximal independent sets—which is originated from Erd?s and Moser—is widely investigated, much less is known in general. In this paper we settle the question for trees and prove that the maximum number of k‐dominating independent sets in n‐vertex graphs is between and if , moreover the maximum number of 2‐dominating independent sets in n‐vertex graphs is between and . Graph constructions containing a large number of k‐dominating independent sets are coming from product graphs, complete bipartite graphs, and finite geometries. The product graph construction is associated with the number of certain Maximum Distance Separable (MDS) codes.     

k-Arcs,Hyperovals, Partial Flocks and Flocks

Some recent results on k-arcs and hyperovals of PG(2,q),on partial flocks and flocks of quadratic cones of PG(3,q),and on line spreads in PG(3,q) are surveyed. Also,there is an appendix on how to use Veronese varieties as toolsin proving theorems.     

On a question of Thas on partial 3‐(qn + 1, q + 1, 1) designs

In this note, we answer a question of JA Thas about partial $3$‐ $\left(q^n+1,q+1,1\right)$ designs. We then extend this answer to a result about the embedding of certain partial $3$‐ $\left(q^2+1,q+1,1\right)$ designs into Möbius planes.     

Strongly regular locally <Emphasis Type="Italic">GQ</Emphasis>(4,<Emphasis Type="Italic">t</Emphasis>)-graphs

Amply regular with parameters (v, k, λ, μ) we call an undirected graph with v vertices in which the degrees of all vertices are equal to k, every edge belongs to λ triangles, and the intersection of the neighborhoods of every pair of vertices at distance 2 contains exactly μ vertices. An amply regular diameter 2 graph is called strongly regular. We prove the nonexistence of amply regular locally GQ(4,t)-graphs with (t,μ) = (4, 10) and (8, 30). This reduces the classification problem for strongly regular locally GQ(4,t)-graphs to studying locally GQ(4, 6)-graphs with parameters (726, 125, 28, 20).     

On the duals of certain d-dimensional dual hyperovals in

Let d2. A construction of d-dimensional dual hyperovals in PG(2d+1,2) using quadratic APN functions was discovered by Yoshiara in [S. Yoshiara, Dimensional dual hyperovals associated with quadratic APN functions, Innov. Incidence Geom., in press]. In this note, we prove that the duals of the d-dimensional dual hyperovals in PG(2d+1,2) constructed from quadratic APN functions are also d-dimensional dual hyperovals in PG(2d+1,2) if, and only if, d is even. Some examples are presented.     

Hyperovals in nets of small degree

The existence of an r‐net of order n with a hyperoval is proved for r = 5 when n ≥ 63; for r = 7 when n ≥ 84; for r = 9 when n ≥ 59,573 and for r = 15 when n ≥ 1, 873,273. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 322–334, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10018     

On d-Dimensional Dual Hyperovals

d-dimensional dual hyperovals in a projective space of dimension n are the natural generalization of dual hyperovals in a projective plane. After proving some general properties of them, we get the classification of two-dimensional dual hyperovals in projective spaces of order 2. A characterization of the only two-dimensional dual hyperoval which is known in PG(5,4) is also given. Finally the classification of 2-transitive two-dimensional dual hyperovals is reached.