C(31,6,2) <=31

Method of construction: cyclic covering found by a search program

 2  3  5 10 14 20
 3  4  6 11 15 21
 4  5  7 12 16 22
 5  6  8 13 17 23
 6  7  9 14 18 24
 7  8 10 15 19 25
 8  9 11 16 20 26
 9 10 12 17 21 27
10 11 13 18 22 28
11 12 14 19 23 29
12 13 15 20 24 30
13 14 16 21 25 31
 1 14 15 17 22 26
 2 15 16 18 23 27
 3 16 17 19 24 28
 4 17 18 20 25 29
 5 18 19 21 26 30
 6 19 20 22 27 31
 1  7 20 21 23 28
 2  8 21 22 24 29
 3  9 22 23 25 30
 4 10 23 24 26 31
 1  5 11 24 25 27
 2  6 12 25 26 28
 3  7 13 26 27 29
 4  8 14 27 28 30
 5  9 15 28 29 31
 1  6 10 16 29 30
 2  7 11 17 30 31
 1  3  8 12 18 31
 1  2  4  9 13 19