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We present results on the size of the smallest maximal partial ovoids and on the size of the smallest maximal partial spreads of the generalized quadrangles W (q) and Q(4, q).

In 1969, Denniston gave a construction of maximal arcs of degree d in Desarguesian projective planes of even order q, for all d dividing q. In 2002 Mathon gave a construction method generalizing the one of Denniston. We will give a new geometric approach to these maximal arcs. This will allow us to count the number of non-isomorphic Mathon maximal arcs of… (More)

- David C Clark, Vladimir Tonchev, Donald Kreher, Stefaan De Winter, Steven Seidel, Mark Gockenbach
- 2015

3 Multi-step majority logic decoding and the modified finite geometry designs. 39

Let Π = (P, L, I) denote a rank two geometry. In this paper, we are interested in the largest value of |X||Y | where X ⊂ P and Y ⊂ L are sets such that (X ×Y)∩I = ∅. Let α(Π) denote this value. We concentrate on the case where P is the point set of PG(n, q) and L is the set of k-spaces in PG(n, q). In the case that Π is the projective plane PG(2, q),… (More)

In this article, we prove that amongst all n by n bipartite graphs of girth at least six, where n = q 2 + q + 1 ≥ 157, the incidence graph of a projective plane of order q, when it exists, has the maximum number of cycles of length eight. This characterizes projective planes as the partial planes with the maximum number of quadrilaterals.

In this note we characterize thick finite generalized quadrangles constructed from a generalized hyperoval as those admitting an abelian Singer group, i.e., an abelian group acting regularly on the points.

An argument concerning the isomorphism of certain digraphs leads to a result concerning the number of roots of two trinomials over a finite field of odd characteristic. In turn, this result is generalised via a short proof to produce an unexpected one-to-one correspondence between the image sets of two classes of binomials over finite fields of any… (More)