Some new resolvable GDDs with k=4 and doubly resolvable GDDs with k=3

A doubly resolvable packing design with block size k, index λ, replication number r, and v elements is called a generalized Kirkman square and denoted by GKSk(v;1,λ;r). Existence of GKS3(4u;1,1;2(u−1))s and GKS3(6u;1,1;3(u−1))s is implied by existence of doubly resolvable group divisible designs with block size 3, index 1, and types 4u and 6u (i.e.,  (3,1)-DRGDDs of types 4u and 6u). In this paper, we establish the spectra of (3,1)-DRGDDs of types 4u and 6u with 15 and 31 possible exceptions, respectively. As applications, we get some new classes of permutation codes and doubly constant weight codes. We also construct 5 new resolvable GDDs with block size 4 and index 1.