Some results concerning {(q+1)(n−1);n}-arcs and {(q+1)(n−1)+1;n}-arcs in finite projective planes of order q

Small Complete Arcs in Projective Planes

In the late 1950’s, B. Segre introduced the fundamental notion of arcs and complete arcs [48, 49]. An arc in a finite projective plane is a set of points with no three on a line and it is complete if cannot be extended without violating this property. Given a projective plane P, determining n(P), the size of its smallest complete arc, has been a major open question in finite geometry for severa...

Concerning maximal arcs and inversive planes

Arcs in Finite Projective Spaces

These notes are an outline of a course on arcs given at the Finite Geometry Summer School, University of Sussex, June 26-30, 2017. Basic objects and definitions Let K denote an arbitrary field. Let Fq denote the finite field with q elements, where q is the power of a prime p. Let Vk(K) denote the k-dimensional vector space over K. Let PGk−1(K) denote the (k − 1)-dimensional projective space ove...

Transitive Arcs in Planes of Even Order

When one considers the hyperovals in PG (2 , q ) , q even , q . 2 , then the hyperoval in PG (2 , 4) and the Lunelli – Sce hyperoval in PG (2 , 16) are the only hyperovals stabilized by a transitive projective group [10] . In both cases , this group is an irreducible group fixing no triangle in the plane of the hyperoval , nor in a cubic extension of that plane . Using Hartley’s classification ...

New arcs in projective Hjelmslev planes over Galois rings

It is known that some good linear codes over a finite ring (R-linear codes) arise from interesting point constellations in certain projective geometries. For example, the expurgated Nordstrom-Robinson code, a nonlinear binary [14, 6, 6]-code which has higher minimum distance than any linear binary [14, 6]-code, can be constructed from a maximal 2-arc in the projective Hjelmslev plane over Z4. W...