HPMDs of type $2^n{3}^1$ with Block Size Four and Related HCOLSs

Abstract

A holey perfect Mendelsohn design of type $h_1^{n_1}{h}_2^{n_2}\dots h_k^{n_k}$ (HPMD$(h_1^{n_1}{h}_2^{n_2}\dots h_k^{n_k})$), with block size four is equivalent to a frame idempotent quasigroup satisfying Stein’s third law with the same type, where a frame quasigroup of type $h_1^{n_1}{h}_2^{n_2}\dots h_k^{n_k}$ means a quasigroup of order $n$ with $n_i$ missing subquasigroups (holes) of order $h_i$, $1\le i\le k$, which are disjoint and spanning, that is $\sum _{1\le i\le k}{n}_i{h}_i=n$. In this paper, we investigate the existence of HPMD$(2^n{3}^1)$ and show that an HPMD$(2^n{3}^1)$ exists if and only if $n\ge 4$. As an application, we readily obtain HSOLS$(2^n{3}^1)$ and establish the existence of $(2,3,1)$ [or $(3,1,2)$]-HCOLS$(2^n{3}^1)$ for all $n\ge 4$.