Computational Verification of Projective-Geometric Structures Over GF(19 K ): Singer Cycles, Complete Arcs, and Baer Subplanes for k = 1, 2

Abstract

Classical constructions in finite projective geometry — Singer cycles, complete arcs, and Baer-subplane partitions — are usually established by pure counting and group-theoretic argument, following the field extension GF( ) GF(q2) In this paper we take a complementary, computational-experimental approach for the prime q = 19: we implement the relevant field and incidence structures directly, and verify by explicit computation rather than by counting argument alone that (i) a Singer cycle of PG(2,19) has order 381 and acts sharply transitively on its points, (ii) the standard conic of PG(2,19) is a complete 20-arc, and (iii) randomized greedy search consistently finds smaller complete arcs of size 11–14, well below the conic bound, illustrating the gap between the maximal and the typical complete arc. We validate the whole computational pipeline, including the Baer-subplane fixed-point test, on the small reference case = 3 (GF(9)), where every step can be checked by hand, before applying the same code to the k = 2 level (GF(192) = GF(361)) to confirm the combinatorial partition of PG(2,361) into 343 disjoint Baer subplanes of order 19. All reported figures for = 19 that carry the label ―computed‖ were produced by direct enumeration in this study; figures labelled ―derived‖ follow from the cited counting theorems and were checked only arithmetically.