On the existence and the nonexistence of some (k,n)‎-arcs in PG (2,37)‎

الملخص EN

A (k, n)−arc is a set of k points of a projective plane such that some n, but no n+1 of them, are collinear.

The maximum size of a (k, n)−arc in PG(2 q) is denoted by mn(2,q).

In this paper we proved that 666 ≤ m21(2,37) ≤ 741, 703 ≤ m22(2,37) ≤ 779, 739 ≤ m23(2,37) ≤ 817, 777 ≤ m24(2,37) ≤ 855, 816 ≤ m25(2,37) ≤ 893, 854 ≤ m26(2,37) ≤ 931, 894 ≤ m27(2,37) ≤ 969, 933 ≤ m28(2,37) ≤ 1007, 970 ≤ m29(2,37) ≤ 1045, m30(2,37) ≤ 1083, m31(2,37) ≤ 1121, m32(2,37) ≤ 1159, m33(2,37) ≤ 1197, m34(2,37) ≤ 1235, m35(2,37) ≤ 1273 and m36(2,37) ≤ 1311.