The asymptotic existence of orthogonal designs

Given any -tuple ( s1, s2, . . . , s ) of positive integers, there is an integer N = N ( s1, s2, . . . , s ) such that an orthogonal design of order 2 ( s1 + s2 + · · ·+ s ) and type ( 2s1, 2 s2, . . . , 2 s ) exists, for each n ≥ N . This complements a result of Eades et al. which in turn implies that if the positive integers s1, s2, . . . , s are all highly divisible by 2, then there is a full orthogonal design of type ( s1, s2, . . . , s ) .