Can't be bothered to read the whole thing at the moment and I'm sure I'm not the only one, so care to explain what's so exciting? Any fun we can have from it?

I didn't quite understand a a lot of the jargon, what I get from it is that with these squares, you can create a tournament that allows each player to play each of the seven countries while making sure that each player shares a border with another player around the same number of times as any other player, making the tournament more balanced.

I think it would be a great idea to implement these into tournament. It prevents the boring outcome of bordering with the same players over and over again. It allows for

disregard those last 3 words, my brain is completely fried right now Dx

What's the point of 1408 as opposed to 1 latin squares?

The point is that is how many permutations there are for a 7 player 7 game tournament where all players border all opponents exactly 4 or 5 times across the 7 games. This is interesting and useful for tournament finals.

Deinodon - You should ask the author if he knows of any similar papers (or ideally matrix solutions) focusing on opening round balance. I.e. A 43 player diplomacy tournament where each player plays each opponent exactly once, since a 7 game round yields 7*6 opponents (42 opponents for player X so 43 total players if you want to play each unique opponent exactly once).