The solutions up to and including n=8 meet the same criteria (a) to (d) from para 2.2 of Fried and Mansour (2023) for the square of the matrices. See the comment in A368539. However, the trace of the optimal matrices is larger than in the case of squared matrices. When n=9, the row and column totals for the same index are no longer the same, but 4 of these totals differ by 2. See linked file.