For what odd integers $n \geq 3$ does there exist a self-complementary $(2n,8n−4,4n−2,n,2n−2)$ balanced incomplete block design?
By "self-complementary" I mean that the complement of each block is a block itself. A (trivial) example of such a design is given by the set of all twenty 3-element subsets of a set of size 6 ($n=3$).
Motivating problem: One wants to divide $2n$ players into 2 equal-sized teams in $4n−2$ different ways in such a way that each player is on the same team as each other player exactly $2n−2$ times. See my earlier MO question Self-complementary block designs , which asked about dividing the players into teams in $2n-1$ ways so that each player is on the same team as each other player exactly $n-1$ times. When $n$ is odd, no such scheme exist, as can be shown by an easy parity argument; here I am asking whether, by doubling the number of games played, one can get around the parity obstacle and obtain a design of the desired kind.