Uniform semi-Latin squares and their pairwise-variance aberrations
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Volume
213
Pagination
282 - 291
Publisher
DOI
10.1016/j.jspi.2020.12.003
Journal
Journal of Statistical Planning and Inference
ISSN
0378-3758
Metadata
Show full item recordAbstract
For integers
and
, an
semi-Latin square is an
array of
-subsets (called blocks) of an
-set (of treatments), such that each treatment occurs once in each row and once in each column of the array. A semi-Latin 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
semi-Latin square is Schur optimal in the class of all
semi-Latin squares, and here we show that when a uniform
semi-Latin square exists, the Schur optimal
semi-Latin squares are precisely the uniform ones. We then compare uniform semi-Latin squares using the criterion of pairwise-variance (PV) aberration, introduced by J. P. Morgan for affine resolvable designs, and determine the uniform
semi-Latin squares with minimum PV aberration when there exist
mutually orthogonal Latin squares of order
. These do not exist when
, and the smallest uniform semi-Latin squares in this case have size
. We present a complete classification of the uniform
semi-Latin squares, and display the one with least PV aberration. We give a construction producing a uniform
semi-Latin square when there exist
mutually orthogonal Latin squares of order
, and determine the PV aberration of such a uniform semi-Latin square. Finally, we describe how certain affine resolvable designs and balanced incomplete-block designs can be constructed from uniform semi-Latin squares. From the uniform
semi-Latin 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 incomplete-block designs for 36 treatments in 84 blocks of size 6. In particular, this shows that there are at least 16875 pairwise non-isomorphic orthogonal arrays
.
Authors
Bailey, RA; Soicher, LHCollections
- Mathematics [1463]