I think the following schedule is about the best that it possible for 15 weeks.
Court 1 Court 2 Byes
( 8 6 v 10 3) ( 5 4 v 7 2) ( 1 9)
( 9 7 v 6 4) ( 1 5 v 8 3) ( 2 10)
(10 8 v 7 5) ( 2 1 v 9 4) ( 3 6)
( 6 9 v 8 1) ( 3 2 v 10 5) ( 4 7)
( 7 10 v 9 2) ( 4 3 v 6 1) ( 5 8)
(10 3 v 4 7) ( 8 9 v 2 5) ( 1 6)
( 6 4 v 5 8) ( 9 10 v 3 1) ( 2 7)
( 7 5 v 1 9) (10 6 v 4 2) ( 3 8)
( 8 1 v 2 10) ( 6 7 v 5 3) ( 4 9)
( 9 2 v 3 6) ( 7 8 v 1 4) ( 5 10)
( 1 10 v 4 5) ( 8 7 v 9 3) ( 6 2)
( 2 6 v 5 1) ( 9 8 v 10 4) ( 7 3)
( 3 7 v 1 2) (10 9 v 6 5) ( 8 4)
( 4 8 v 2 3) ( 6 10 v 7 1) ( 9 5)
( 5 9 v 3 4) ( 7 6 v 8 2) (10 1)
Where players have exactly one bye in each block of 5 weeks. This means that everyone will get exactly 12 games, which unfortunately must give an unbalanced schedule. Each player has 12 partners, and at best will partner 6 players once and the other 3 players twice. Also, each player has 24 opponents, and at best will oppose 3 players twice and the other 6 players three times. The schedule above does meet these two criteria, as well as having optimal social mixing, since all pairs of players play together, either as partners or opponents, exactly 4 times.
Hope that helps.
