Holey Schröder designs of type 3nu1

A holey Schröder design of type h1n1h4n2⋯hknk (HSD(h1n1h4n2⋯hknk)) is equivalent to a frame idempotent Schröder quasigroup (FISQ(h1n1h4n2⋯hknk)) of order n with ni missing subquasigroups (holes) of order hi,1≤i≤k, which are disjoint and spanning (i.e., ∑1≤i≤knihi=n). The existence of an HSD(hn) is completely solved, and the existence of HSD(hnu1) for h=1,2, and 4 has been known with a few of possible exceptions. In this paper, we consider the existence of HSD(3nu1). It is shown that for 1≤u≤15 and u≠3, an HSD(3nu1) exists if and only if n(n+2u−1)≡0(mod4), n≥4, and n≥1+2u/3. For 1≤u≤n and u≠3, an HSD(3nu1) exists if and only if n(n+2u−1)≡0(mod4) and n≥4, with possible exceptions of n=29,43. We have also found six new HSDs of type (4nu1).