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Abstract
For integers n > 2 and k > 0, an (n×n)∕k semiLatin square is an n × n array of ksubsets (called blocks) of an nkset (of treatments), such that each treatment occurs once in each row and once in each column of the array. A semiLatin square is uniform if every pair of blocks, not in the same row or column, intersect in the same positive number of treatments. It is known that a uniform (n × n)∕k semiLatin square is Schur optimal in the class of all (n × n)∕k semiLatin squares, and here we show that when a uniform (n × n)∕k semiLatin square exists, the Schur optimal (n × n)∕k semiLatin squares are precisely the uniform ones. We then compare uniform semiLatin squares using the criterion of pairwisevariance (PV) aberration, introduced by J. P. Morgan for affine resolvable designs, and determine the uniform (n × n)∕k semiLatin squares with minimum PV aberration when there exist n−1 mutually orthogonal Latin squares of order n. These do not exist when n=6, and the smallest uniform semiLatin squares in this case have size (6 × 6)∕10. We present a complete classification of the uniform (6 × 6)∕10 semiLatin squares, and display the one with least PV aberration. We give a construction producing a uniform ((n + 1) × (n + 1)) ∕ ((n − 2)n) semiLatin square when there exist n − 1 mutually orthogonal Latin squares of order n, and determine the PV aberration of such a uniform semiLatin square. Finally, we describe how certain affine resolvable designs and balanced incompleteblock designs can be constructed from uniform semiLatin squares. From the uniform (6 × 6)∕10 semiLatin squares we classified, we obtain (up to block design isomorphism) exactly 16875 affine resolvable designs for 72 treatments in 36 blocks of size 12 and 8615 balanced incompleteblock designs for 36 treatments in 84 blocks of size 6. In particular, this shows that there are at least 16875 pairwise nonisomorphic orthogonal arrays OA (72,6,6,2).
Original language  English 

Pages (fromto)  282291 
Journal  Journal of Statistical Planning and Inference 
Volume  213 
Early online date  19 Dec 2020 
DOIs  
Publication status  Published  Jul 2021 
Keywords
 Design optimality
 Block design
 Schur optimality
 Affine resolvable design
 Balanced incompleteblock design
 Orthogonal array
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Dive into the research topics of 'Uniform semiLatin squares and their pairwisevariance aberrations'. Together they form a unique fingerprint.Projects
 1 Finished

CoDiMa: CoDiMa (CCP in the area of Computational Discrete Mathematics)
Linton, S. A. (PI) & Konovalov, O. (CoI)
1/03/15 → 29/02/20
Project: Standard