42-arcs in PG(2, q) left invariant by PSL(2, 7)

Abstract

For an odd prime p ? 7, let q be a power of p such that $${q^3\equiv1 \pmod 7}$$ . It is known that the desarguesian projective plane PG(2, q) of order q has a unique conjugacy class of projectivity groups isomorphic to PSL(2, 7). For such a projective group Γ, we investigate the geometric properties of the (unique) Γ-orbit ? of size 42 such that the 1-point stabilizer of Γ in ? is a cyclic group of order 4. We present a computational approach to prove that ? is a 42-arc provided that q ? 53 and q ? 373, 116, 56, 36. We discuss the case q = 53 in more detail showing the completeness of ? for q = 53.