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Showing papers in "Journal of Geometry in 2000"


Journal ArticleDOI
TL;DR: In this article, the symmetric (3, 3)-nets are found as subnets of affine resolvable 2-(27, 9, 4, 4) designs.
Abstract: Symmetric nets are affine resolvable designs whose duals are also affine. It is shown that. up to isomorphism, there are exactly four symmetric (3, 3)-nets (v=b=27,k=9), and exactly two inequivalent 9×9 generalized Hadamard matrices over the group of order 3. The symmetric (3, 3)-nets are found as subnets of affine resolvable 2-(27, 9, 4) designs. Ten of the 68 non-isomorphic affine resolvable 2-(27, 9, 4) designs are not extensions of symmetric (3, 3)-subnets, providing the first examples of affine 2-(q3, q2, q2−1/q−1) designs without symmetric (q, q)-subnets.

32 citations


Journal ArticleDOI
TL;DR: A computer search method for finding small complete caps inPG(d,q) is considered, formulated as a combinatorial optimization problem to which record-to-record travel, a stochastic local search method, is applied.
Abstract: A computer search method for finding small complete caps inPG(d,q) is considered. The problem is formulated as a combinatorial optimization problem to which record-to-record travel, a stochastic local search method, is applied. Several new complete caps are found and upper bounds on the number of points in the smallest complete caps inPG(d,q) are tabulated for small parameters.

26 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that no semioval can contain a full line, and that apart from two small cases, no semovals can contain all but one point of some line.
Abstract: Asemioval in a projective plane Π is a setS of points such that for every pointP e S, there exists a unique line l of Π such thatl∩S={P}. In other words, at every point ofS, there exists a unique tangent line. In this paper, we consider semiovals such that some line has a “large” intersection withS. In a finite plane Π it is shown that no semioval can contain a full line, and that apart from two small cases, no semioval can contain all but one point of some line. We then consider semiovals which contain all but two points of some line, providing some examples and characterizations.

18 citations


Journal ArticleDOI
TL;DR: The finite translation planes of ordern not in {34,36,112,192,292,592} that admit a non-solvable doubly transitive line-sized orbit are completely classified as mentioned in this paper.
Abstract: The finite translation planes of ordern not in {34,36,112,192,292,592 that admit a non-solvable doubly transitive line-sized orbit are completely classified

16 citations


Journal ArticleDOI
TL;DR: In this paper, the results of a number of computer searches in projective planes of order 16 were presented, showing that maximal arcs of degree 4 can be found in all but two of the known planes and their duals, yielding a resolvable 2-(52, 4, 1) design with at least 52 resolutions.
Abstract: This paper tabulates the results of a number of computer searches in projective planes of order 16. Maximal arcs of degree 4 are found in all but two of the known planes of order 16 (and their duals). Any such arc yields a resolvable 2-(52, 4, 1) design that admits at least 52 resolutions. Pairs of disjoint degree 4 maximal arcs are also shown to exist in certain of the planes giving rise to 104-sets of type (4, 8).

16 citations


Journal ArticleDOI
TL;DR: An interesting class of k-arcs with k = 4(√−1) in the projective plane over GF(q) is constructed for q an odd square; the construction yields many complete arcs of small size in PG(2,q) when q≤2401.
Abstract: An interesting class ofk-arcs withk=4(√−1) in the projective plane overGF(q) is constructed forq an odd square; the construction yields many complete arcs of small size inPG(2,q) whenq≤2401.

15 citations


Journal ArticleDOI
TL;DR: In this paper, a unified approach to the Blaschke-Lebesgue Theorem and the Firey-Sallee Theorem on Reuleaux polygons in the Euclidean plane is presented.
Abstract: We give a unifying approach to the Blaschke-Lebesgue Theorem and the Firey-Sallee Theorem on Reuleaux polygons in the Euclidean plane.

14 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that various conformal groups, including classical conformal diffeomorphism groups, are essential and that their essentiality is equivalent to the non-vanishing of a conformal cohomological invariant.
Abstract: We show that various conformal groups, including classical conformal diffeomorphism groups are essential. The essentiality is shown to be equivalent to the non-vanishing of a conformal cohomological invariant.

14 citations


Journal ArticleDOI
TL;DR: In this paper it was shown that if a plane of PG(3,q),q even, meets an ovoid in a pointed conic, then either q = 4 and the ovoid is an elliptic quadric, or q = 8 and the OV is a Tits ovoid.
Abstract: It is shown that if a plane of PG(3,q),q even, meets an ovoid in a pointed conic, then eitherq=4 and the ovoid is an elliptic quadric, orq=8 and the ovoid is a Tits ovoid.

12 citations



Journal ArticleDOI
TL;DR: In this paper, a synthetic construction of the irregular hyperoval has been proposed, and the construction is used to determine the full group of automorphisms of the hyperoval, provide computer-free proofs of its known properties and introduce some new results concerning the intersection of this hyperoval and conics in the given plane.
Abstract: This irregular hyperoval, found by computer search in 1958, has many interesting and unusual properties. We provide a synthetic construction of this hyperoval, use the construction to determine the full group of automorphisms of the hyperoval, provide computer-free proofs of its known properties and introduce some new results concerning the intersection of this hyperoval and conics in the given plane.

Journal ArticleDOI
TL;DR: In this paper, the authors give the answer of a question by H. Lenz, wether for 3 ≥k
Abstract: We give the answer of a question by H. Lenz, wether for 3≤k

Journal ArticleDOI
TL;DR: In this article, the smallest blocking sets with respect to higher dimensional subspaces in the quadrics Q(2n, q) and Q+(2n+ 1, q).
Abstract: We determine the three smallest blocking sets with respect to lines of the quadric Q(2n, q) withn ≥ 3 and the two smallest blocking sets with respect to lines of the quadric Q+(2n+1,q) withn ≥ 2. These results will be used in a forthcoming paper for determining the smallest blocking sets with respect to higher dimensional subspaces in the quadrics Q(2n, q) and Q+(2n+ 1, q).

Journal ArticleDOI
TL;DR: The first 5-(72, 6, 1) designs with automorphism group PSL(2, 71) were enumerated in this article, where a necessary and sufficient condition for semiregular 5-(v, 6.1, v 1)-designs to exist is that v = 84, 228 (mod 360).
Abstract: The first 5-(72, 6, 1) designs with automorphism group PSL(2, 71) were found by Mills [10]. We presently enumerate all 5-(72, 6, 1) designs with this automorphism group. There are in all 926299 non-isomorphic designs. We show that a necessary condition for semiregular5-(v, 6, 1) designs with automorphism group PSL(2, v 1) to exist is thatv=84, 228 (mod 360). In particular, there are exactly 3 non-isomorphic semiregular 5-(84, 6, 1) designs with automorphism group PSL(2, 83). There are at least 6450 non-isomorphic 5-(244, 6, 1) designs with automorphism group P∑L(2, 35).

Journal ArticleDOI
TL;DR: In this article, the integrability of the horizontal distribution of an almost-Kahler or nearly-kahler submersion is studied and curvature properties of such submersions are investigated.
Abstract: In this paper the integrability of the horizontal distribution of an almost-Kahler or a nearly-Kahler submersion is studied and curvature properties of such submersions are investigated.

Journal ArticleDOI
TL;DR: In this paper, it was shown that every plane quadrangulation G has at least one *-orientation and that any two orientations of G can be transformed into one another by a sequence of 4-cycle reversals.
Abstract: Aplane quadrangulation G is a simple plane graph such that each face ofG is quadrilateral. A (*) -orientation D*(G) ofG is an orientation ofG such that the outdegree of each vertex on ∂G is 1 and the outdegrees of other vertices are all 2, where ∂G denotes the outer 4-cycle ofG. In this paper, we shall show that every plane quadrangulationG has at least one (*)-orientation. We also show that any two (*)-orientations ofG can be transformed into one another by a sequence of 4-cycle reversals. Moreover, we apply this fact toorthogonal plane partitions, which are partitions of a square into rectangles by straight segments.

Journal ArticleDOI
TL;DR: For any odd integern ≥ 3 and prime power q ≥ 3, it is known that for any odd integral n−1, q2, the Singular Cycle can be partitioned into pairwise disjoint subgeometries isomorphic to the Singer cycle of P(n−1 and q2) by taking point orbits under an appropriate subgroup of a Singer cycle as discussed by the authors.
Abstract: For any odd integern ≥3 and prime powerq, it is known thatPG(n−1, q2) can be partitioned into pairwise disjoint subgeometries isomorphic toPG(n−1, q) by taking point orbits under an appropriate subgroup of a Singer cycle ofPG(n−1, q2). In this paper, we construct Baer subgeometry partitions of these spaces which do not arise in the classical manner. We further illustrate some of the connections between Baer subgeometry partitions and several other areas of combinatorial interest, most notably projective sets and flagtransitive translation planes.


Journal ArticleDOI
TL;DR: In this article, the dual of a generalized hexagon Γ of finite order (s, t) is associated to the Chevalley groups if and only if the intersection of any two tracesxy andxz, with some additional condition, contains at mostt/s + 1 elements.
Abstract: We characterize the dual of the generalized hexagons naturally associated to the groupsG2(q) and3D4(q) by looking at certain configurations, and also by considering intersections of traces. For instance, the dual of a generalized hexagon Γ of finite order (s, t) is associated to the Chevalley groups mentioned above if and only if the intersection of any two tracesxy andxz, with some additional condition, contains at mostt/s + 1 elements.

Journal ArticleDOI
TL;DR: In this article, the authors characterize affine translation surfaces with constant Gaussian curvature and show that such surfaces must be flat and that one of the defining curves must be a planar curve.
Abstract: In this paper we characterize affine translation surfaces with constant Gaussian curvature. We show that such surfaces must be flat and that one of the defining curves must be planar.

Journal ArticleDOI
TL;DR: In this paper, the stability of euclidean balls with one pinched curvature measure is proved giving an explicit stability order which depends only on the dimension of the ambient space, which is meant with respect to the Hausdorff measure for compact convex bodies.
Abstract: In the main result the stability of euclidean balls with one pinched curvature measure is proved giving an explicit stability order which depends only on the dimension of the ambient space. Stability here is meant with respect to the Hausdorff measure for compact convex bodies. The technique of the proof involves generalized Minkowski integral formulas, inner parallel bodies and estimates for the isoperimetric defect. The result also improves stability estimates for orthogonal disc cylinders in the noncompact case.

Journal ArticleDOI
TL;DR: In this paper, it was shown that in projective spaces of dimension three, the two plane characterk-sets forke {q 2+ 1,(q+1)2} are of the same type as the elliptic or the hyperbolic quadric, respectively.
Abstract: In this paper we prove that in a projective space of dimension three and orderq the two plane characterk-sets forke {q 2+ 1,(q+1)2} are of the same type as the elliptic or the hyperbolic quadric, respectively. As a corollary we obtain a characterization of the elliptic and the hyperbolic quadrics.

Journal ArticleDOI
TL;DR: In this paper, it was shown that some of these graphs are the point graph of exactly one partial geometry, i.e., the triality quadric Q+(7,2).
Abstract: Up to now there are eight partial geometries pg(7,8,4) known. Their point graphs as well as their block graphs are all related to the triality quadric Q+(7,2). We prove that some of these graphs are the point graph of (up to isomorphism) exactly one partial geometry. We investigate the relations among some of these eight partial geometries. Generalizing our results, we construct two new families of partial geometries pg(22n−1− 1, 22n−1, 22n−2).

Journal ArticleDOI
TL;DR: In this article, the Klein-Kroll types of Minkowski planes over half-ordered fields are determined with respect to G- and q-translations and (p, q)-homotheties.
Abstract: This paper concerns a construction of Minkowski planes over half-ordered fields [5] and [20]. Solving various functional equations the Klein-Kroll types of these Minkowski planes are determined with respect toG- andq-translations and (p, q)-homotheties. Examples for some of the resulting types are given.

Journal ArticleDOI
TL;DR: In this paper, it is shown that, by applying hyperbolic isometries, one may arrange points in hyper-bolic (n−1)-space in a certain standard position.
Abstract: It is shown that, by applying hyperbolic isometries, one may arrangen points in hyperbolic (n−1)-space in a certain standard position. Similar results are developed for complex hyperbolic space, as well as hyperbolic spaces defined over nearly arbitrary fields. The algebraic basis of the paper is the determination of the structure of a double coset space which occurs in representation theory.

Journal ArticleDOI
TL;DR: In this paper, recursive constructions for simple 3-designs have been proposed, which are generalization of several recent results obtained by the author and prove to be very useful.
Abstract: This paper concerns recursive constructions for simple 3-designs. These are generalization of several recent results obtained by the author. The methods are based on 3-designs having a parallelism and prove to be very useful. Illustrative applications are included to demonstrate their surprising strength and as a result new infinite families of 3-designs are constructed.

Journal ArticleDOI
Mario Delanote1
TL;DR: In this paper, it was shown that under the condition of existence of an orthogonal spread, the graph INQ(2m, 3),m ≥ 3, is the point graph of a new semipartial geometry SPQ(1, 2), where vertices are adjacent when non-orthogonal.
Abstract: Consider the graph INQ(2m, 3) on the internal points of a quadric Q(2m, 3), where vertices are adjacent when non-orthogonal. We prove that form odd and under the condition of existence of an orthogonal spread, the graph INQ(2m, 3),m ≥3, is the point graph of a new semipartial geometry SPQ(2m, 3).

Journal ArticleDOI
TL;DR: This article established connections between Desarguesian partial parallelisms of deficiency one in PG(3,q) and translation planes of orderq4 admitting a collineation group isomorphic to SL(2,q), which is generated by Baer collineations.
Abstract: This article establishes connections between Desarguesian partial parallelisms of deficiency one in PG(3,q) and translation planes of orderq4 admitting a collineation group isomorphic to SL(2,q) which is generated by Baer collineations.

Journal ArticleDOI
Bart De Bruyn1
TL;DR: In this article, an upper bound for the cardinality of a set of points in PG(n, q) with the property that nol of them are contained in a (l− 2)-fiat (n ≥l − 2 ≥ 0) was derived.
Abstract: We find an upper bound for the cardinality of a set of points in PG(n, q) with the property that nol of them are contained in a (l− 2)-fiat (n ≥l − 2 ≥ 0) and we treat the case of equality. We also determine all ovoids and Cameron closed sets of the regular near-hexagon related to the extended ternary Golay code.

Journal ArticleDOI
TL;DR: In this paper, an explicit formula for the dimensions of k-nuclei is given for #F=F≥k+1, where F is the number of non-zero digits in the representation ofn+1 in basep.
Abstract: Ak-nucleus of a normal rational curve inPG(n, F) is the intersection over allk-dimensional osculating subspaces of the curve (k e {−1,0,...,n− 1}). It is well known that for characteristic zero all nuclei are empty. In case of characteristicp}>0 and #F≥n the number of non-zero digits in the representation ofn+ 1 in basep equals the number of distinct nuclei. An explicit formula for the dimensions ofk-nuclei is given for #F=F≥k+ 1.