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 GKS k ( v ; 1 , λ ; r ) . Existence of GKS 3 ( 4 u ; 1 , 1 ; 2 ( u - 1 ) ) s and GKS 3 ( 6 u ; 1 , 1 ; 3 ( u - 1 ) ) s is implied by existence of doubly resolvable group divisible designs with block size 3, index 1, and types 4 u and 6 u (i.e.,? ( 3 , 1 ) -DRGDDs of types 4 u and 6 u ). In this paper, we establish the spectra of ( 3 , 1 ) -DRGDDs of types 4 u and 6 u 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.