12 research outputs found
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
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 there exists a unique line t_P such that . 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
Some Blocking Semiovals which Admit a Homology Group
AbstractThe study of blocking semiovals in finite projective planes was motivated by Batten in connection with cryptography. Dover in studied blocking semiovals in a finite projective plane of order q which meet some line inq− 1 points. In this note, some blocking semiovals in PG(2, q) are considered which admit a homology group, and three new families of blocking semiovals are constructed. Any blocking semioval in the first or the third family meets no line in q− 1 points
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 , 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 /
2-semiarcs in PG(2, q), q <= 13
A 2-semiarc is a pointset S-2 with the property that the number of tangent lines to S-2 at each of its points is two. Using some theoretical results and computer aided search, the complete classification of 2-semiarcs in PG(2, q) is given for q <= 7, the spectrum of their sizes is determined for q <= 9, and some results about the existence are proven for q = 11 and q = 13. For several sizes of 2-semiarcs in PG(2, q), q <= 7, classification results have been obtained by theoretical proofs