A generalization of perfect Mendelsohn designs and some of its consequences

A generalization of Mendelsohn designs to the case of repetition of elements in their blocks has been made. The notion of perfect, twice strongly solvable generalized Mendelsohn designs has been introduced. It has been shown that the construction of these designs in case $\lambda=k$ (DSR-$(v,k,k)$-PRMD) can be reduced to the construction of a special matrix of size $v\times k$ based on elements of the set $X=\{1,2,\dots,v\}$. Connections of DSR-$(v,k,k)$-PRMD with the following combinatorial constructions has been shown: systems of $k$ pairwise orthogonal $F$-rectangular schemes of size $v\times kv$; systems of $k$ pairwise orthogonal $F$-squares of order $kv$; rectangular orthogonal schemes of size $(k+2)\times kv^2$ and of weight $(v-1)vk$; solvable BIB$(vk;v^2k,vk,k,k)$-designs; solvable designs GDD$(vk,k,v,0,k)$. This connection has been illustrated by the example of DSR-$(3,5,5)$-PRMD.