A Lower Bound on Blocking Semiovals

Abstract

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.

Two Families of Blocking Semiovals

“…The first one is a family of blocking semiovals of size 3q − 4 in P G (2, q), where q ≥ 5. This families contains the family constructed by Dover [4] as a special case. The second one is a family of blocking semiovals of size 3q e − q − 2 in P G(2, q e ), where q ≥ 3 and e ≥ 2.…”

“…Clearly x 0 = 0 and x 1 = |S| by the definition of S. Dover [5] proved that x q = 0 for q > 3. Vertexless triangles and the blocking semiovals constructed by Dover [4] have the property x q−1 = 0. Set X (S) = (x 1 , x 2 , .…”

“…Then 2q + 2 ≤ |S| ≤ q √ q + 1. The lower bound was proved by Dover [4] and the upper bound was proved by Hubaut [6]. The upper bound is met if and only if S is a unital [2].…”

“…Dover says that the only families of blocking semiovals that seem to be known are unitals, vertexless triangles, and the family given by [4…”

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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.

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