Holey Schröder designs of type 4n u1

A holey Schröder design of type h1 1 h n2 2 . . . h nk k (HSD(h n1 1 h n2 2 . . . h nk k )) is equivalent to a frame idempotent Schröder quasigroup (FISQ(h1 1 h n2 2 . . . hk k )) of order n with ni missing subquasigroups (holes) of order hi, 1 ≤ i ≤ k, which are disjoint and spanning, that is, ∑1≤i≤k nihi = n. In this paper, we consider the existence of HSD(4u) for 0 ≤ u ≤ 36 and show that these HSDs exist if and only if 0 ≤ u ≤ 2n − 2 and n ≥ 4 with just nine possible exceptions. We also investigate the existence of HSD(4u) for general u and prove that there exists an HSD(4u) for u ≥ 37 and n ≥ 2u/3 + 7.