Abstract
Let V be a finite set of v elements. A packing of the pairs of V by k-subsets is a family F of k-subsets of V, called blocks, such that each pair in V occurs in at most one member of F. For fixed v and k, the packing problem is to determine the number of blocks in any maximum packing. A maximum packing is resolvable if we can partition the blocks into classes (called parallel classes) such that every element is contained in precisely one block of each class. A resolvable maximum packing of the pairs of V by k-subsets is denoted by RP(v,k). It is well known that an RP(v,4) is equivalent to a resolvable group divisible design (RGDD) with block 4 and group size h, where h=1,2 or 3. The existence of 4-RGDDs with group-type h n for h=1 or 3 has been solved except for (h,n)=(3,4) (for which no such design exists) and possibly for (h,n)∈{(3,88),(3,124)}. In this paper, we first complete the case for h=3 by direct constructions. Then, we start the investigation for the existence of 4-RGDDs of type 2n. We shall show that the necessary conditions for the existence of a 4-RGDD of type 2n, namely, n ≥ 4 and n ≡ 4 (mod 6) are also sufficient with 2 definite exceptions (n=4,10) and 18 possible exceptions with n=346 being the largest. As a consequence, we have proved that there exists an RP(v,4) for v≡ 0 (mod 4) with 3 exceptions (v=8,12 or 20) and 18 possible exceptions.
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Gennian Ge Researcher supported in part by YNSFC Grant 10001026
C.W.H. Lam Researcher supported by the National Science and Research Council of Canada
Alan C.H. Ling Researcher supported by an ARO grant 19-01-1-0406 and a DOE grant
Researcher supported by NSFC Grant 19831050
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GE, G., LAM, C.W.H., Ling, A.C.H. et al. Resolvable Maximum Packings with Quadruples. Des Codes Crypt 35, 287–302 (2005). https://doi.org/10.1007/s10623-003-6739-6
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DOI: https://doi.org/10.1007/s10623-003-6739-6