I had not seen plan2play before so I took a quick look, and I ended up with a schedule where one of the 20 players had 9 others players who they never played with either as partners or opponents. It's possible to do a lot lot better and I offer the schedule below which gives you everything that you asked for except for the fact that players 15 & 19 oppose twice.
(10 3 v 7 14) ( 9 11 v 2 1) (13 15 v 5 12) ( 8 6 v 4 16)
(17 5 v 19 9) ( 4 11 v 7 10) (12 1 v 3 6) ( 2 18 v 20 8)
( 7 20 v 17 6) ( 8 14 v 19 1) ( 4 5 v 16 18) (15 3 v 2 13)
(16 12 v 9 2) ( 1 3 v 18 17) ( 4 19 v 15 11) (14 10 v 20 13)
(10 12 v 8 18) ( 9 6 v 14 19) (20 11 v 5 15) (17 7 v 16 13)
( 9 8 v 15 7) (12 13 v 1 4) ( 5 2 v 10 6) ( 3 14 v 16 11)
(17 11 v 6 12) ( 5 20 v 8 1) (10 4 v 9 3) ( 7 18 v 19 2)
(17 3 v 8 5) (15 18 v 14 6) (13 19 v 2 16) ( 7 4 v 20 1)
(17 2 v 4 14) (15 1 v 16 10) ( 3 12 v 20 19) ( 9 18 v 13 11)
(12 14 v 5 7) (20 18 v 9 16) (11 6 v 13 8) (17 19 v 10 15)
The byes have a pattern to them 17,18,19,20 in round 1, 13,14,15,16 in round 2, etc.. Hope that helps.