On extendable planes of order ten (Q762159)

The author proves that if L is a quasi-symmetric \(2-(n^ 2+n+1,n+2,n)\)- design (for the definition of quasi-symmetric, see \textit{P. J. Cameron} and \textit{J. H. van Lint} [Graphs, codes and designs (1980; Zbl 0427.05001) p. 25], then there is a projective plane \(\pi\) of order n such that the blocks of L form a class of hyperovals of \(\pi\) and the lines of \(\pi\) augmented by a new point, together with all the blocks of L, form an extension of \(\pi\). It follows that a projective plane of order 10 (if there is one) has an extension if and only if there is a quasi-symmetric 2-(111,12,10)-design.