C(12,4,2) <=12

Method of construction: cyclic covering found by a search program

 2  3  5  9
 3  4  6 10
 4  5  7 11
 5  6  8 12
 1  6  7  9
 2  7  8 10
 3  8  9 11
 4  9 10 12
 1  5 10 11
 2  6 11 12
 1  3  7 12
 1  2  4  8