There are only so many combinations of rankings that are possible. It shouldn't be hard to encode each of those possible orders and transmit them as a whole.
O(n^2) isn't great though...
The algorithm to count is rough and requires many rounds (since you count, knock out the lowest candidate, recount, and repeat until a >50% favor is achieved by one candidate). Cardinal systems on the other hand just require summing the columns and taking argmax. This is far simpler (in fact we can do most of this in parallel making a far better runtime).
With 20 candidates in an election, that's over 2 billion billion possibilities. It's not practical.