Here's one way to play the 1st 15 weeks. All pairs of players are together within a foursome exactly 4 times, and each player has one bye in each of the three 5 week blocks. There is a small issue with the same three players together in consecutive weeks in the 1st block.
Time 1 Time 2 Bye
( 1 6 4 10) ( 5 8 7 2) (3 9)
( 2 7 5 6) ( 1 9 8 3) (4 10)
( 3 8 1 7) ( 2 10 9 4) (5 6)
( 4 9 2 8) ( 3 6 10 5) (1 7)
( 5 10 3 9) ( 4 7 6 1) (2 8)
( 6 8 3 9) ( 2 1 10 5) (4 7)
( 7 9 4 10) ( 3 2 6 1) (5 8)
( 8 10 5 6) ( 4 3 7 2) (1 9)
( 9 6 1 7) ( 5 4 8 3) (2 10)
(10 7 2 8) ( 1 5 9 4) (3 6)
( 1 3 10 2) ( 6 9 5 7) (4 8)
( 2 4 6 3) ( 7 10 1 8) (5 9)
( 3 5 7 4) ( 8 6 2 9) (1 10)
( 4 1 8 5) ( 9 7 3 10) (2 6)
( 5 2 9 1) (10 8 4 6) (3 7)
For the last 15 weeks, use the same schedule, with a different assignment of player names to the numbers 1 to 10.