Resolvable Maximum Packings with Quadruples

Author： GE Gennian LAM C. Ling Alan Shen Hao

ISSN： 0925-1022

Source： Designs, Codes and Cryptography, Vol.35, Iss.3, 2005-06, pp. : 287-302

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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 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.