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Holey Schröder designs of type 2 n u 1
Author(s) -
Bennett F. E.,
Wei Ruizhong,
Zhang Hantao
Publication year - 1998
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/(sici)1520-6610(1998)6:2<131::aid-jcd4>3.0.co;2-g
Subject(s) - quasigroup , mathematics , disjoint sets , combinatorics , idempotence , order (exchange) , type (biology) , ecology , finance , economics , biology
A holey Schröder design of type $h^{n_{1}}_{1}h^{n_{2}}_{2} \cdots h^{n_{k}}_{k}$ $[HSD(h^{n_{1}}_{1}h^{n_{2}}_{2} \cdots h^{n_{k}}_{k})]$ is equivalent to a frame idempotent Schröder quasigroup $[FISQ(h^{n_{1}}_{1}h^{n_{2}}_{2} \cdots h^{n_{k}}_{k})] of order m with n i missing subquasigroups (holes) of order h i , 1 ≤ i ≤ k , which are disjoint and spanning, that is, ∑ 1≤ i ≤ k n i h i = m . In this article, we first consider the existence of HSD(2 n u 1 ) for 1 ≤ u ≤ 4 and show that these HSDs exist if and only if n ≥ u + 1 with the exception of ( n , u ) ∈ {(2, 1), (3, 1), (3, 2)}. Then we investigate the existence of HSD(2 n u 1 ) for general u and prove that there exists an HSD(2 n u 1 ) for u ≥ 16 and n ≥ [5 u /4] + 14. © 1998 John Wiley & Sons, Inc. J Combin Designs 6:131–150, 1998