Construction from K6 graph factorization The relations between the graph factors of the complete graph K6 generate an S(5,6,12). A K6 graph has 6 vertices, 15 edges, 15 perfect matchings, and 6 different 1-factorizations (ways to partition the edges into disjoint perfect matchings). The set of vertices (labeled 123456) and the set of factorizations (labeled ABCDEF) provide one block each. Every pair of factorizations has exactly one perfect matching in common. Suppose factorizations A and B have the common matching with edges 12, 34 and 56. Add three new blocks AB3456, 12AB56, and 1234AB, replacing each edge in the common matching with the factorization labels in turn. Similarly add three more blocks 12CDEF, 34CDEF, and 56CDEF, replacing the factorization labels by the corresponding edge labels of the common matching. Do this for all 15 pairs of factorizations to add 90 new blocks. Finally, take the full set of combinations of 6 objects out of 12, and discard any combination that has 5 or more objects in common with any of the 92 blocks generated so far. Exactly 40 blocks remain, resulting in blocks of the S(5,6,12). This method works because there is an outer automorphism on the symmetric group S6, which maps the vertices to factorizations and the edges to partitions. Permuting the vertices causes the factorizations to permute differently, in accordance with the outer automorphism.
Construction from K6 graph factorization The relations between the graph factors of the complete graph K6 generate an S(5612) A K6 graph has 6 vertices 15 edges 15 perfect matchings and 6 different 1-factorizations (ways to partition the edges into disjoint perfect matchings) The set of vertices (labeled 123456) and the set of factorizations (labeled ABCDEF) provide one block each Every pair of factorizations has exactly one perfect matching in common Suppose factorizations A and B have the common matching with edges 12 34 and 56 Add three new blocks AB3456 12AB56 and 1234AB replacing each edge in the common matching with the factorization labels in turn Similarly add three more blocks 12CDEF 34CDEF and 56CDEF replacing the factorization labels by the corresponding edge labels of the common matching Do this for all 15 pairs of factorizations to add 90 new blocks Finally take the full set of combinations of 6 objects out of 12 and discard any combination that has 5 or more objects in common with any of the 92 blocks generated so far Exactly 40 blocks remain resulting in blocks of the S(5612) This method works because there is an outer automorphism on the symmetric group S6 which maps the vertices to factorizations and the edges to partitions Permuting the vertices causes the factorizations to permute differently in accordance with the outer automorphism