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Further results on large sets of partitioned incomplete Latin squares
Author(s) -
Shen Cong,
Li Dongliang,
Cao Haitao
Publication year - 2021
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/jcd.21772
Subject(s) - mathematics , combinatorics , latin square , prime (order theory) , existential quantification , spectrum (functional analysis) , discrete mathematics , rumen , chemistry , food science , quantum mechanics , fermentation , physics
In this article, we continue to study the existence of large sets of partitioned incomplete Latin squares (LSPILS). We complete the determination of the spectrum of an LSPILS ( g n ) and prove that there exists an LSPILS ( g n ) if and only if g ≥ 1 , n ≥ 3 , and ( g , n ) ≠ ( 1 , 6 ) . We also start the investigation of LSPILS with two group sizes and prove that there exists an LSPILS + ( g n( 2 g ) 1 ) for all g ≥ 1 and n ≥ 3 except possibly for n ≡ 2 , 10( mod 12 ) and n ≥ 14 . Furthermore, we obtain a pair of orthogonal LSPILS + ( 1 p 2 1 ) s, where p is a prime and p ≡ 1( mod 6 ) .