A Classification of Hyperfocused 12-Arcs

A k-arc in PG(2, q) is a set of k points no three which are collinear. hyperfocused the $$k \atopwithdelims ()2$$ secants meet some external line exactly $$k-1$$ points. Hyperfocused k-arcs can be viewed as 1-factorizations complete graph $$K_k$$ that embed q). We study 526,915,620 $$K_{12}$$ , determine embeddable q), and classify 12-arcs. Specifically we show if 12-arc $$\mathcal {K}$$ arc then $$q = 2^{5k}$$ subset hyperconic including nucleus.