Designs With Mutually Orthogonal Resolutions

A combinatorial design D with replication number r is said to be resolvable if the blocks of D can be partitioned into classes R 1 , R 2 , . . . , R r such that each element of D is contained in precisely one block of each class. Two resolutions R and R ′ of D are called orthogonal if | R i ⋂ R ′ j | ⩽ 1 for all R j ∈ R , R ′ j ∈ R ′. A set Q = { R 1 , R 2 , . . . , R t } of t resolutions of D is called a set of mutually orthogonal resolutions (MORs) if the resolutions of Q are pairwise orthogonal. We are interested in the existence of designs with sets of t mutually orthogonal resolutions for t ⩾ 2 and in determining bounds for t . Constructions are given for sets of t MORs for t ⩾ 2 for some group divisible designs and ( r , λ)-designs. We then consider the problem of determining the size of a largest set of MORs for a design. The best known bounds for t are presented and discussed.