Resolvable maximum packings with quadruples

• Authors: Gennian Ge;C. W. H. Lam;Alan C. H. Ling;Hao Shen
• Venue: Designs, Codes and Cryptography
• Year: 2005

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 hn 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.