Holey Perfect Mendelsohn Designs of Type 2 n u 1 with Block Size Four
Hantao Zhang
Source 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 2 n 3 1 , was established by Bennett et al. in 1997. In this paper, we investigate the existence of 4-HPMDs of type 2 n u 1 for 1 ≤ u ≤ 16 : a 4-HPMD( 2 n u 1 ) 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( 2 n u 1 ) for general u and prove that there exists a 4-HPMD( 2 n u 1 ) for all n ≥ ⌈ 5 u / 4 ⌉ + 4 . Moreover, if u ≥ 35 , then a 4-HPMD( 2 n u 1 ) exists for all n ≥ ⌈ 5 u / 4 ⌉ + 1 ; if u ≥ 95 , then a 4-HPMD( 2 n u 1 ) exists for all n ≥ ⌈ 5 u / 4 ⌉ − 2 .
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