A survey on semiovals

A semioval in a finite projective plane is a non-empty pointset S with the property that for every point in $S$ there exists a unique line t_P such that $S \cap t_P = {P}$. This line is called the tangent to S at P. Semiovals arise in several parts of finite geometries: as absolute points of a polarity (ovals, unitals), as special minimal blocking sets (vertexless triangle), in connection with cryptography (determining sets). We survey the results on semiovals and give some new proofs

Blocking semiovals of Type (1,M+1,N+1)

We consider the existence of blocking semiovals in finite projective planes which have intersection sizes 1, m+1 or n+1 with the lines of the plane for 1 \leq m < n. For those prime powers $q \leq 1024$, in almost all cases, we are able to show that, apart from a trivial example, no such blocking semioval exists in a projective plane of order q. We are also able to prove, for general q, that if q2+q+1 is a prime or three times a prime, then only the same trivial example can exist in a projective plane of order q. <br /

Two Families of Blocking Semiovals

AbstractThe study of blocking semiovals in finite projective planes was motivated by Batten in connection with cryptography and was begun by Dover . In this note, two new families of blocking semiovals are constructed in desarguesian planes