Steiner triple systems of order 19 associated with a certain type of projective plane of order 10 (Q1819728)

Let a hypothetical projective plane of order 10 be given through its incidence matrix, where rows correspond to lines and columns correspond to points; assume that the binary code generated by the rows of this matrix contains a vector of weight 16; this vector yields a configuration in the plane consisting of a set of 16 points and 8 ''special'' lines, each containing 4 points of the set. Under a further assumption on this configuration the author proves that it is possible to single out 19 lines of the plane and form 57 triples of these lines, yielding a Steiner triple system enjoying three special properties, which he calls (P1), (P2), (P3). On the other side, the existence of a Steiner triple system on 19 elements satisfying (P1), (P2), (P3) would lead to the completion of a large portion of the incidence matrix of a projective plane of order 10. The author then analyzes the known Steiner triple systems on 19 elements to see if they satisfy (P1), (P2), (P3); using a particular \(v\) to 2\(v+1\) construction for Steiner triple systems, one can start from the unique Steiner triple system on 9 elements and get, up to isomorphism, three Steiner triple systems on 19 elements [cf. the author, Proc. Roy. Soc. Edinb., Sect. A 94, 90-92 (1983; Zbl 0514.05017)]: in each of these three systems either (P1) or (P2) fails; in another example taken from the class of reverse Steiner triple systems [cf. \textit{J. Doyen}, Discrete Math. 1, 315-319 (1972; Zbl 0272.05013)] property (P2) fails again; the author has started the exhaustive investigation of the class of such reverse Steiner triple systems on 19 elements, which contains, up to isomorphism, 184 candidates.