Orthogonal systems

An equidistant permutation array (E.P.A.)A(r, λ v) is av × r array in which every row is a permutation of the integers 1, 2, ⋯,r such that any two distinct rows have precisely λ columns in common. In this paper we introduce the concept of orthogonality for E.P.A.s. A special case of this is the well known idea of a set of pairwise orthogonal latin squares. We show that a set of these arrays is equivalent to a particular type of resolvable (r, λ)-design. It is also shown that the cardinality of such a set is bounded byr − λ with the upper bound being obtained only ifλ = 0. A brief survey of related orthogonal systems is included. In particular, sets of pairwise orthogonal symmetric latin squares, sets of orthogonal Steiner systems and sets of orthogonal skeins.