Resolvable packings R˜MP(3,2;n,n-3) and coverings R˜MC(3,2;n,n-2)

Let n≡k-1,0n≡k-1,0 or 1(modk). An R˜MP(k,λ;n,m) (resp. R˜MC(k,λ;n,m)) is a resolvable packing (resp. covering) with maximum (resp. minimum) possible number m   of parallel classes which are mutually distinct, each parallel class consists of ⌊(n-k+1)/k⌋⌊(n-k+1)/k⌋ blocks of size k   and one block of size n-k⌊(n-k+1)/k⌋n-k⌊(n-k+1)/k⌋, and its leave (resp. excess) is a simple graph. Such designs can be used to construct certain uniform designs which have been widely applied in industry, system engineering, pharmaceutics, and natural sciences. In this paper, direct and recursive constructions are discussed for such designs. The existence of an R˜MP(3,2;n,n-3) and an R˜MC(3,2;n,n-2) for n≡1(mod3) is established with n⩾16n⩾16.