On putative \(q\)-analogues of the Fano plane and related combinatorial structures (Q2832883)

The authors study sets of planes in \(6\)-dimensional projective space over the finite field \({\mathbb F}_q\) with the property that every line is contained in precisely one plane. This is a special instance of a \textit{design over a finite field} in the sense of \textit{P. J. Cameron} [in: Combinatorics, Proc. British combin. Conf., Aberystwyth 1973, 9--13 (1974; Zbl 0298.05018)]. The dimensions in the involved spaces in the underlying vector-space model are two, three and seven, respectively, hence the name ``\(q\)-analogue of the Fano plane''. (Recall that the Fano plane is a Steiner system \(S(2,3,7)\).) While some general constructions for such designs over finite fields do exist, many question remain open. They are interesting because of their relation to subspace codes. In fact, Steiner systems over finite fields give rise to optimal subspace codes.NEWLINENEWLINEThis paper reports on the authors' attempts to construct \(q\)-analogous of the Fano plane. It actually fails and existence of such \(q\)-analogous remains an open problem (as indicated by the word ``putative'' in the title). However, a lot of new insight is gained and promising machinery is developed that may well enable at least partial answers to the existence question in the future. As an important by-product, new plane subspace codes of record size in projective space of dimension six over \({\mathbb F}_q\) are found. Starting from a general subspace code of size \(q^8+q^5+q^4-q-1\) whose planes meet a fixed \(3\)-flat \(S\), certain plains are removed and the resulting subcode is extended again by planes that meet \(S\) in a line. This results in plane subspace codes of size \(329\) for \(q = 2\) (without computer aid!) and size \(6977\) for \(q = 3\).NEWLINENEWLINEFor the entire collection see [Zbl 1343.00043].