Graeco-Latin squares with embedded balanced superimpositions of Youden squares
✍ Scribed by D.A. Preece; B.J. Vowden
- Publisher
- Elsevier Science
- Year
- 1995
- Tongue
- English
- Weight
- 468 KB
- Volume
- 138
- Category
- Article
- ISSN
- 0012-365X
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✦ Synopsis
Parker's (1959)
first example of a 10 x 10 Graeco-Latin square incorporates 4 balanced superimpositions of 3 x 7 Youden squares. Such superimpositions of size s x (2s + 1), where s is odd and (2s + 1) is prime, can be of two types, distinguished by the values taken by an invariant formed from the incidence matrices for the superimpositions. All four of the superimpositions in Parker's example are of Type 1. We now give 10 x 10 Graeco Latin squares similar to Parker's, but with x ( = 0, 2, 3) superimpositions of Type 1 and 4 -x of Type 2. Our 10 x 10 examples with x = 0, 2, 3 and 4 are shown to be special cases of constructions for Graeco-Latin squares of order 3s + 1; Graeco-Latin squares with x = 1 are shown to be impossible.