Minimum \((12, 6, 3)\) Covers

A \((12,6,3)\) cover is a family of 6-element subsets, called blocks, chosen from a 12-element universe, such that each 3-element subset is contained in at least one block. This paper constructs a \((12,6,3)\) cover with 15 blocks, and it shows that any \((12,6,3)\) cover has at least 15 blocks; thus the covering number \(C(12,6,3) = 15\). It also shows that the 68 nonisomorphic \((12,6,3)\) covers with 15 blocks fall into just two classes using a very natural classification scheme.