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 Ω of size 42 such that the 1-point stabilizer of Γ in Ω is a cyclic group of order 4. We present a computational approach to prove that Ω 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 Ω for q = 53.
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This is one of several papers published together in Designs, Codes and Cryptography on the special topic: “Geometry, Combinatorial Designs & Cryptology”.
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Indaco, L., Korchmáros, G. 42-arcs in PG(2, q) left invariant by PSL(2, 7). Des. Codes Cryptogr. 64, 33–46 (2012). https://doi.org/10.1007/s10623-011-9532-y
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DOI: https://doi.org/10.1007/s10623-011-9532-y