Some Existence on Ordered Multi-designs

Two variants of an orthogonal array, orthogonal arrays of type I and of type II, were introduced by Rao in 1961. Furthermore, as generalizations of an orthogonal array and an orthogonal array of type II, an orthogonal multi-array and a perpendicular multi-array have been introduced by Brickell in 1984 and by Li et al. in 2018, respectively. In this paper, as a generalization of the orthogonal array of type I, an ordered multi-design is newly introduced from a combinatorial viewpoint. Necessary conditions for the existence of an ordered multidesign are discussed and several constructions of the ordered multi-design are provided by use of group divisible designs and self-orthogonal latin squares, through a difference technique. As main results, the existence of a family of ordered multi-designs is provided and also the sufficiency of necessary conditions for existence is shown for a class of ordered multi-designs with one possible exception.