Update: Some new results on lower bounds on $(n,r)$-arcs in $PG(2,q)$ for $q\le 31$

Abstract

An (n, r)-arc in PG(2, q) is a set B of points in PG(2, q) such that each line in PG(2, q) contains at most r elements of B and such that there is at least one line containing exactly r elements of B. The value m r (2, q) denotes the maximal number n of points in the projective geometry PG(2, q) for which an (n, r)-arc exists. By explicitly constructing (n, r)-arcs using prescribed automorphisms and integer linear programming we obtain some improved lower bounds for m r (2, q): m 10 (2, 16) ≥ 144, m 3 (2, 25) ≥ 39, m 18 (2, 25) ≥ 418, m 9 (2, 27) ≥ 201, m 14 (2, 29) ≥ 364, m 25 (2, 29) ≥ 697, m 25 (2, 31) ≥ 734. Furthermore, we show by systematically excluding possible automorphisms that putative (44, 5)-arcs, (90, 9)-arcs in PG(2, 11), and (39, 4)-arcs in PG(2, 13)-in case of their existence-are rigid, i.e. they all would only admit the trivial automorphism group of order 1. In addition, putative (50, 5)-arcs, (65, 6)-arcs, (119, 10)-arcs, (133, 11)-arcs, and (146, 12)-arcs in PG(2, 13) would be rigid or would admit a unique automorphism group (up to conjugation) of order 2.