Intersection numbers for subspace designs

Intersection numbers for subspace designs are introduced and $q$-analogs of the Mendelsohn and K\"ohler equations are given. As an application, we are able to determine the intersection structure of a putative $q$-analog of the Fano plane for any prime power $q$. It is shown that its existence implies the existence of a $2$-$(7,3,q^4)_q$ subspace design. Furthermore, several simplified or alternative proofs concerning intersection numbers of ordinary block designs are discussed

Decompositions of Complete Symmetric Directed Graphs into the Oriented Heptagons

The complete symmetric directed graph of order $v$, denoted $K_{v}^*$, is the directed graph on $v$ vertices that contains both arcs $(x,y)$ and $(y,x)$ for each pair of distinct vertices $x$ and $y$. For a given directed graph, $D$, the set of all $v$ for which $K_{v}^*$ admits a $D$-decomposition is called the spectrum of $D$. There are 10 non-isomorphic orientations of a $7$-cycle (heptagon). In this paper, we completely settled the spectrum problem for each of the oriented heptagons.Comment: 10 pages, 1 figur

Types of directed triple systems

We introduce three types of directed triple systems. Two of these, Mendelsohn directed triple systems and Latin directed triple systems, have previously appeared in the literature but we prove further results about them. The third type, which we call skewed directed triple systems, is new and we determine the existence spectrum to be v ≡ 1 (mod 3), v ≠ 7, except possibly for v = 22, as well as giving enumeration results for small orders

Distributive and anti-distributive Mendelsohn triple systems

We prove that the existence spectrum of Mendelsohn triple systems whose associated quasigroups satisfy distributivity corresponds to the Loeschian numbers, and provide some enumeration results. We do this by considering a description of the quasigroups in terms of commutative Moufang loops. In addition we provide constructions of Mendelsohn quasigroups that fail distributivity for as many combinations of elements as possible. These systems are analogues of Hall triple systems and anti-mitre Steiner triple systems respectively

Row-column directed block designs

AbstractA balanced incomplete block design (BIBD) is called a row-column directed BIBD (RCDBIBD) if: 1.(i) it is directed in the usual sense, i.e., each ordered pair of points occurs an equal number of times in the blocks (directed column wise), and2.(ii) the blocks are arranged in such a way that (a) each point occurs an equal number of times in each row, and (b) each ordered pair of distinct points occurs an almost equal number of times in the rows.The present paper gives construction techniques for RCDBIBDs and proves that the necessary conditions are sufficient for the existence of RCDBIBDs with block size 2. Existence of RCDBIBDs with block size 3 and v ≡ 1 mod 6 is shown and it is proved that an RCDBIBD(7,7,4,4,1∗) does not exist