Telecommunication and incidence structures (Q909233)

The author considers an incidence structure \(S=(V,B,\in)\) with point set V and line set B such that: any two points are on exactly one line; each line contains 3 or 4 points; every point belongs to exactly one 4-line; V is partitioned into n subsets \(V_ 1,...,V_ n\) of 12 points each, and each \(V_ i\) contains 3 pairwise disjoint 4-lines and 4 pairwise disjoint 3-lines (and possibly some additively 3-lines). The existence of such a structure for \(n=2\) was asked by telecommunication engineers. The author gives a construction for a certain S, with arbitrary n, starting by modifying \textit{R. Peltesohn}'s solution [Compos. Math. 6, 251-257 (1938; Zbl 0020.04902)] to a problem by \textit{L. Heffter} [Math. Ann. 49, 101-112 (1897; JFM 28.0128.02)]. For \(n=2\) the corresponding solution is explicitly listed, and is worthwhile to mention that no additional 3-line occurs in \(V_ 1\) or \(V_ 2\), i.e. \(V_ 1\) and \(V_ 2\) are joined with the maximal possible number of 3-lines (72). The article ends with some outlooks and further questions.