The Nonexistence of 4-(12,6,6) Designs

With the help of computer algorithms we prove that there are no 4-(12, 6, 6) designs, thereby answering the last open existence question in design theory for at most 12 points. We also enumerate three families of related designs, namely the 10977 simple 3-(10, 4, 3) designs, the 67 simple 4-(11, 5, 3) designs, and the 23 simple 5-(12, 6, 3) designs. Finally, we complete the census of all possible partitions of 6-sets on 12 points into 5-(12, 6, A) designs and of 5-sets on 11 points into 4-(11, 5, A) designs.