Improved Bounds on mr(2,q)‎ q=19,25,27

Abstract EN

An (n,r)-arc is a set of n points of a projective plane such that some r, but no r+1 of them, are collinear.

The maximum size of an (n,r)-arc in PG(2, q) is denoted by mr(2, q).

In this paper, a new (286, 16)-arc in PG(2,19), a new (341, 15)-arc, and a (388, 17)-arc in PG(2,25) are constructed, as well as a (394, 16)-arc, a (501, 20)-arc, and a (532, 21)-arc in PG(2,27).

Tables with lower and upper bounds on mr(2, 25) and mr(2, 27) are presented as well.

The results are obtained by nonexhaustive local computer search.