Holey Frames With Block Size Three

Frames with some special properties are powerful for the construction of other typesof combinatorial designs in design theory. A holey frame is a holey group divisible designwhose blocks can be partitioned into holey parallel classes, each of which is with respectto some group. Holey frames can be used to construct some other resolvable designs suchas RGDD，IRGDD and frames. So, it is very valuable to discuss the existence of holeyframe.In this article, we prove that the necessary conditions for a holey frame with blocksize3are also sufficient with some possible exceptions. There exists a (3,λ)-HF (n, m~t)if and only if n≥4, t≥3, λm (t-1)≡0(mod2), and mt (n-1)≡0(mod3), exceptfor (λ,n,m,t)=(1,6,1,3) and possibly except for (1)(λ,n,m,t)=(2i+1,6,1,6j+3),i≥0, j≥1;(2) λ=1,(m, t)=(2,6), n∈{8,10,12,14,15,18};(3)λ=1, n=6,(m, t)∈{(4,6),(3,7)}. Our paper is given with five chapters as follows:In Chapter1, some concepts are introduced, such as group divisible design, holeyframe and so on. And, the necessary condition for a holey frame are listed.In Chapter2, we give some recursive constructions for holey frames.In Chapter3, some direct constructions are listed.In Chapter4, we obtain our main conclusions.In Chapter5, we list some Further problems.