Is There an Algorithm for Solving the Combinatorial Design Problem?

[Constantinos]

Hey!

I have a certain problem. Let M ≥ 4 be an even number and consider the set [0,1,...$\frac{M}{2}$-1]. The problem is to put those numbers two times in each row of an M x (M choose 2) matrix, such that all possible combinations of entries that contain a pair of the same number occur just once.

For example, M = 4 it can be trivially seen that the matrix will be:

[0 0 1 1;
0 1 0 1;
0 1 1 0;
1 0 0 1;
1 0 1 0;
1 1 0 0]

Indeed all possible combinations of entries that contain 0 in a row, occur just once. This is also true for all possible combinations of entries that contain the number 1.

For M = 6 though, things get much more difficult. Is there an algorithm that can produce such matrices for arbitrary M? Does such a matrix even exist? Any papers or other info? Thanks!

 

[fresh_42]

You simply have an $\frac{M}{2}-$ary representation of numbers with certain values of their norm, defined as sum of the digits.