Two Families of Blocking Semiovals

Suetake, Chihiro

doi:10.1006/eujc.2000.0402

Cited by 14 publications

(7 citation statements)

References 5 publications

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“…(This example belongs to a class of blocking semiovals which was first described by Suetake [7].) The line X 1 = X 2 is a (q − 1)-secant of S, but the tangent to S at the point P a = (a, a, 1) has equation X 1 = aX 3 or X 2 = aX 3 according to a is a square or a non-square element in GF(q) * .…”

Section: Small Semiovals With Large Collinear Subsetsmentioning

confidence: 99%

Small semiovals in PG(2, q)

Kiss

2008

J. geom.

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A semioval in a projective plane is a nonempty subset S of points with the property that for every point P ∈ S there exists a unique line such that S ∩ = {P }. It is known that q + 1 ≤ |S| ≤ q √ q + 1 and both bounds are sharp. We say that S is a small semioval in if |S| ≤ 3(q + 1).Dover [5] proved that if S has a (q − 1)-secant, then 2q − 2 ≤ |S| ≤ 3q − 3, thus S is small, and if S has more than one (q − 1)-secant, then S can be obtained from a vertexless triangle by removing some subset of points from one side. We generalize this result and prove that if there exist integers 1 ≤ t and −1 ≤ k such that |S| = 2q − t + k, 2(t + k) < q, t + 4(k + 1) < q and S has a (q − t)-secant, then the tangent lines at the points of the (q − t)-secant are concurrent. Specially when t = 1 then S can be obtained from a vertexless triangle by removing some subset of points from one side. (2000): 51E21. Mathematics Subject Classification

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Section: Small Semiovals With Large Collinear Subsetsmentioning

confidence: 99%

Small semiovals in PG(2, q)

Kiss

2008

J. geom.

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“…2. A family of size 3q e − q − 2 in P G(2, q e ) where q is a prime power with q ≥ 3, also discovered by Suetake [9]. 3.…”

Section: Resultsmentioning

confidence: 95%

A Lower Bound on Blocking Semiovals

Dover

2000

European Journal of Combinatorics

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A semioval in a projective plane is a set S of points such that for every point P ∈ S, there exists a unique line of such that ∩ S = {P}. In other words, at every point of S, there exists a unique tangent line. A blocking set in is a set B of points such that every line of contains at least one point of B, but is not entirely contained in B. Combining these notions, we obtain the concept of a blocking semioval, a set of points in a projective plane which is both a semioval and a blocking set. Batten [1] indicated applications of such sets to cryptography, which motivates their study. In this paper, we give some lower bounds on the size of a blocking semioval, and discuss the sharpness of these bounds.

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“…Bounds on the sizes of blocking semiovals, as well as a number of constructions have appeared in the literature (see [4], [12], [15], [14]). While the definition of a blocking semioval is valid in any projective plane, we will restrict ourselves to consideration of the finite Desarguesian plane over the field GF(q) for q a prime power, denoted by PG(2, q).…”

Section: Introductionmentioning

confidence: 99%

Blocking semiovals containing conics

Dover

Mellinger

Wantz

2013

Advances in Geometry

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A blocking semioval is a set of points in a projective plane that is both a blocking set (i.e., every line meets the set, but the set contains no line) and a semioval (i.e., there is a unique tangent line at each point). Szőnyi investigated an infinite family of blocking semiovals that are formed from the union of conics contained in a particular type of algebraic pencil. In this paper, the authors look at the general problem of blocking semiovals containing conics, proving a lower bound on the size of such sets and providing several new constructions of blocking semiovals containing conics. In addition, the authors investigate the natural generalization of Szőnyi's construction to other conic pencils.

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Two Families of Blocking Semiovals

Abstract: The study of blocking semiovals in finite projective planes was motivated by Batten [1] in connection with cryptography and was begun by Dover [4,5]. In this note, two new families of blocking semiovals are constructed in desarguesian planes.

Cited by 14 publications

References 5 publications

Small semiovals in PG(2, q)

Small semiovals in PG(2, q)

A Lower Bound on Blocking Semiovals

Blocking semiovals containing conics

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