Complete enumeration and properties of binary pseudo-Youden designs PYD(9,6,6)

A binary pseudo-Youden design PYD(9,6,6) is a 6×6 array in which each cell contains 1 element from the set V={1,2,…,9}, and each element from V occurs 4 times. Every row of the array contains distinct elements and every column contains distinct elements. The rows and columns, when taken together, are pairwise balanced and form a (9,12,8,6,5)-BIBD. In Preece [1968. Balanced 6×6 designs for 9 treatments. Sankhyā Ser. B 30, 443-446; 1976. A second domain of balanced 6×6 designs for nine equally-replicated treatments. Sankhyā Ser. B 38 (part 2) 192-194] a total of 345 species of binary PYD(9,6,6) were found. Here we complete this enumeration and find 348 species of binary PYD(9,6,6). Thus we find 3 new species. We give a complete set of invariants for these species based upon the numbers of intercalates and anti-intercalates that they contain; and discuss some of their properties. We also show that there are 696 non-isomorphic binary PYD(9,6,6), and give a complete set of invariants for these arrays.