On Hyperfocused Arcs in PG(2,q)

Abstract

A k-arc in a Dearguesian projective plane whose secants meet some external line in k-1 points is said to be hyperfocused. Hyperfocused arcs are investigated in connection with a secret sharing scheme based on geometry due to Simmons. In this paper it is shown that point orbits under suitable groups of elations are hyperfocused arcs with the significant property of being contained neither in a hyperoval, nor in a proper subplane. Also, the concept of generalized hyperfocused arc, i.e. an arc whose secants admit a blocking set of minimum size, is introduced: a construction method is provided, together with the classification for size up to 10