Holey Perfect Mendelsohn Designs of Type 2nu1 with Block Size Four

Abstract

Let 4-HPMD denote a holey perfect Mendelsohn design with block size four. The existence of 4-HPMDs with n holes of size 2 and one hole of size 3, that is, of type 2n31, was established by Bennett et al. in 1997. In this paper, we investigate the existence of 4-HPMDs of type 2nu1 for 1≤u≤16: a 4-HPMD(2nu1) exists if and only if n≥max(4,u+1), except possibly for (n,u)=(7,5), (7, 6), (11, 9), (11, 10). We also investigate the existence of 4-HPMD(2nu1) for general u and prove that there exists a 4-HPMD(2nu1) for all n≥⌈5u/4⌉+4. Moreover, if u≥35, then a 4-HPMD(2nu1) exists for all n≥⌈5u/4⌉+1; if u≥95, then a 4-HPMD(2nu1) exists for all n≥⌈5u/4⌉−2.