HPMDs of type \(2^n3^1\) with Block Size Four and Related HCOLSs

A holey perfect Mendelsohn design of type \(h_1^{n_1} h_2^{n_2} \ldots h_k^{n_k}\) (HPMD\((h_1^{n_1} h_2^{n_2} \ldots h_k^{n_k})\)), 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 \(h_1^{n_1} h_2^{n_2} \ldots h_k^{n_k}\) means a quasigroup of order \(n\) with \(n_i\) missing subquasigroups (holes) of order \(h_i\), \(1 \leq i \leq k\),
which are disjoint and spanning, that is \(\sum_{1\leq i \leq k} n_ih_i = n\). In this paper, we investigate the existence of HPMD\((2^n3^1)\) and show that an HPMD\((2^n3^1)\) exists if and only if \(n \geq 4\). As an application, we readily obtain HSOLS\((2^n3^1)\) and establish the existence of \((2,3,1)\) [or \((3,1,2)\)]-HCOLS\((2^n3^1)\) for all \(n \geq 4\).