Resolvable maximum packings with quadruples

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 2(n). 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.