On Holey Schréder Designs of Type \(2^n u^1\)

Abstract

It has been shown by Bennett et al. in 1998 that a holey Schröder design with \(n\) holes of size 2 and one hole of size \(u\), i.e., of type \(2^n u\), exists if \(1 \leq u \leq 4\) and \(n \geq u+1\) with the exception of \((n,u) \in \{(2, 1), (3, 1), (3, 2)\}\), or \(u \geq 16\) and \(n \geq \left\lceil \frac{5u}{4} \right\rceil + 14\). In this paper, we extend this result by showing that, for \(1 \leq u \leq 16\), a holey Schröder design of type \(2^n u\) exists if and only if \(n \geq u+1\), with the exception of \((n,u) \in \{(2, 1), (3, 1), (3, 2)\}\) and with the possible exception of \((n,u) \in \{(7,5), (7,6), (11,9), (11,10)\}\). For general \(u\), we prove that there exists an HSD(\(2^n u\)) for all \(u \geq 17\) and \(n \geq \left\lceil \frac{5u}{4} \right\rceil + 4\). Moreover, if \(u \geq 35\), then an HSD(\(2^n u\)) exists for all \(n \geq \left\lceil \frac{5u}{4} \right\rceil + 1\); if \(u \geq 95\), then an HSD(\(2^n u\)) exists for all \(n \geq \left\lceil \frac{5u}{4} \right\rceil – 2\). We also improve a well-known result on the existence of holey Schröder designs of type \(h^n\) by removing the remaining possible exception of type \(64\).