Room square construction (Howell design)

Question

Using superimposed orthogonal Latin squares to construct a balanced tournament design, I get an n x n array of unique ordered pairs. If I only want unique unordered pairs, I can eliminate half of the "broken" diagonals (the diagonals other than the main diagonal), replacing their content with empty cells, to get a Room square or Howell design.

For example, in a square of 7 x 7, I find that the diagonals parallel to the main diagonal that start in columns 1, 2, and 4 contain the same set of pairs as the diagonals that start in columns 6, 5, and 3, respectively, but with the symbols reversed, (x,y) (y,x). I can eliminate the cells in either of those two sets of diagonals and preserve the property that each symbol appears only once in each row and in each column. Intuitively, I suspect that this pattern is a result of the fact that I transposed my Latin square to obtain its orthogonal Latin square.

In this example, I used (6i + 2j) mod 7 as my composition operator and obtained the following juxtaposed orthogonal Latin squares.

  M  |  0    1    2    3    4    5    6  
-----+-----------------------------------
  0  | 1,-  3,7  5,6  7,5  2,4  4,3  6,2 
  1  | 7,3  2,-  4,1  6,7  1,6  3,5  5,4 
  2  | 6,5  1,4  3,-  5,2  7,1  2,7  4,6 
  3  | 5,7  7,6  2,5  4,-  6,3  1,2  3,1 
  4  | 4,2  6,1  1,7  3,6  5,-  7,4  2,3 
  5  | 3,4  5,3  7,2  2,1  4,7  6,-  1,5 
  6  | 2,6  4,5  6,4  1,3  3,2  5,1  7,- 

Eliminating diagonals 3, 5, and 6 yields a Room square. Alternatively, diagonals 1, 2, and 4 could have been eliminated.

  M  |  0    1    2    3    4    5    6  
-----+-----------------------------------
  0  | 1,-  3,7  5,6       2,4           
  1  |      2,-  4,1  6,7       3,5      
  2  |           3,-  5,2  7,1       4,6 
  3  | 5,7            4,-  6,3  1,2      
  4  |      6,1            5,-  7,4  2,3 
  5  | 3,4       7,2            6,-  1,5 
  6  | 2,6  4,5       1,3            7,- 

For relatively small squares, it's not difficult to choose which diagonals to eliminate by inspection and a bit of trial-and-error. However, I'm not comfortable with that as a solution. Is there a construction that will directly show which cells to eliminate?

[This isn't a homework question; I'm not a student.]

Answer 1

The 2nd table is essentially a starter-adder construction of a Room square and so you need to choose the diagonals so that the pairs and their position in row zero meet the defined properties of starter and adder respectively. I believe the literature on starter-adders is going to give you more Room square constructions than you can get starting from pairs of orthogonal Latin squares.