The existence of Schröder designs with equal-sized holes Original Research Article

A holey Schröder design of type h1n1h2n2 … hknk (HSD(h1n1h2n2 … hknk)) is equivalent to a frame idempotent Schröder quasigroup (FISQ(h1n1h2n2 … hknk)) of order n with ni missing subquasigroups (holes) of order hi, (1 ⩽ i ⩽ k), which are disjoint and spanning, that is, Σ1 ⩽ i ⩽ k nihi = n. In this paper, it is shown that an HSD(hn) exists if and only if h2n(n − 1)  0 (mod 4) with expceptions (h, n) ϵ {{(1,5),(1,9),(2,4)}} and the possible exception of (h, n) = (6,4).