Arcs, blocking sets, and minihypers

Abstract A ( k , n )-arc in a finite projective plane П q of order q is a set of k points with some n but no n + 1 collinear points where k > n and 2 ≤ n ≤ q . The maximum value of k for which a ( k , n )-arc exists in PG (2, q ) is denoted by m n (2, q ). It is well known that if n is not a divisor of q , then m n (2, q ) ≤ ( n − 1) q + n − 3. The purpose of this paper is to improve this upper bound on m n (2, q ) using the nonexistence of some minihypers in PG (2, q ) and to characterize some minihypers in PG ( t , q ) where t ≥ 3.