50 research outputs found

    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

    On the number of k-dominating independent sets

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    We study the existence and the number of kk-dominating independent sets in certain graph families. While the case k=1k=1 namely the case of maximal independent sets - which is originated from Erd\H{o}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 kk-dominating independent sets in nn-vertex graphs is between ck22knc_k\cdot\sqrt[2k]{2}^n and ck2k+1nc_k'\cdot\sqrt[k+1]{2}^n if k2k\geq 2, moreover the maximum number of 22-dominating independent sets in nn-vertex graphs is between c1.22nc\cdot 1.22^n and c1.246nc'\cdot1.246^n. Graph constructions containing a large number of kk-dominating independent sets are coming from product graphs, complete bipartite graphs and with finite geometries. The product graph construction is associated with the number of certain MDS codes.Comment: 13 page

    A study of (x(q+1),x;2,q)-minihypers

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    In this paper, we study the weighted (x(q + 1), x; 2, q)-minihypers. These are weighted sets of x(q + 1) points in PG(2, q) intersecting every line in at least x points. We investigate the decomposability of these minihypers, and define a switching construction which associates to an (x(q + 1), x; 2, q)-minihyper, with x <= q(2) - q, not decomposable in the sum of another minihyper and a line, a (j (q + 1), j; 2, q)-minihyper, where j = q(2) - q-x, again not decomposable into the sum of another minihyper and a line. We also characterize particular (x(q + 1), x; 2, q)-minihypers, and give new examples. Additionally, we show that (x(q + 1), x; 2, q)-minihypers can be described as rational sums of lines. In this way, this work continues the research on (x(q + 1), x; 2, q)-minihypers by Hill and Ward (Des Codes Cryptogr 44: 169-196, 2007), giving further results on these minihypers

    Perp-systems and partial geometries

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    A perp-system R(r) is a maximal set of r-dimensional subspaces of PG(N,q) equipped with a polarity rho, such that the tangent space of an element of R(r) does not intersect any element of R(r). We prove that a perp-system yields partial geometries, strongly regular graphs, two-weight codes, maximal arcs and k-ovoids. We also give some examples, one of them yielding a new pg(8,20,2)

    On the structure of the directions not determined by a large affine point set

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    Given a point set UU in an nn-dimensional affine space of size qn1εq^{n-1}-\varepsilon, we obtain information on the structure of the set of directions that are not determined by UU, and we describe an application in the theory of partial ovoids of certain partial geometries

    A geometric approach to Mathon maximal arcs

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