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Aspects of complete sets of \(9\times 9\) pairwise orthogonal latin squares. (English) Zbl 0889.05026

Authors’ abstract: An affine plane of order 9 can be specified by an orthogonal array with 10 constraints and 9 levels. A complete set of pairwise orthogonal \(9 \times 9\) latin squares is obtained when any two of the constraints are taken as rows and columns. Any 3 of the 10 constraints give rise to an adjugacy set of \(9 \times 9\) latin squares from a particular species. For each of the 7 affine planes of order 9 we count the occurrences of different species amongst the 120 subsets of 3 constraints. We give some properties of these species, including the orders of their automorphism groups. We verify the numbers of subplanes of order 2 in each of the 4 projective planes of order 9.

MSC:

05B15 Orthogonal arrays, Latin squares, Room squares
05B25 Combinatorial aspects of finite geometries

Software:

nauty
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References:

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