Holey Schröder Designs of Type $3^{\mathbf{n}}{\mathbf{u}}^1$

Abstract

A holey Schröder design of type $h_1^{n_1}{h}_4^{n_2}\cdots h_k^{n_k}$ (HSD$(h_1^{n_1}{h}_4^{n_2}\cdots h_k^{n_k}))$ is equivalent to a frame idempotent Schröder quasigroup (FISQ$(h_1^{n_1}{h}_4^{n_2}\cdots h_k^{n_k}))$ of order $n$ with $n_i$ missing subquasigroups (holes) of order $h_i,1\le i\le k$, which are disjoint and spanning (i.e., ${\sum }_{1\le i\le k}{n}_i{h}_i=n$). The existence of HSD$(h^n{u}^1)$ for $h=1,2,4$ has been known. In this paper, we consider the existence of HSD$(3^n{u}^1)$ and show that for $0\le u\le 15$, an HSD$(3^n{u}^1)$ exists if and only if $n(n+2u-1)\equiv 0(mod4)$, $n\ge 4$ and $n\ge 1+2u/3$. For $0\le u\le n$, an HSD$(3^n{u}^1)$ exists if and only if $n(n+2u-1)\equiv 0(mod4)$ and $n\ge 4$, with possible exceptions of $n=29,43$. We have also found six new HSDs of type $(4^n{u}^1)$.

26 pages