Enumeration of E ( s 2 ) -Optimal and Minimax-Optimal Supersaturated Designs With 12 Rows, 11 q Columns and s max = 4

Abstract

The $E\left(s^2\right)$-optimal and minimax-optimal supersaturated designs (SSDs) with 12 rows, $11q$ columns, and $s_{\max }=4$ are enumerated in a computer search: there are, respectively, 34, 146, 0, 3, and 1 such designs for $q=2,3,4,5$, and 6. Cheng and Tang proved that for $q>6$, there are no such SSDs. This completes the enumeration of all SSDs with the described restrictions. These results are obtained by enumerating the resolvable 2-($12,6,5q$) designs such that any two blocks not in the same parallel class intersect in 2, 3, or 4 points, and the enumeration is carried out with a breadth-first search algorithm over parallel classes with an isomorph rejection. The combinatorial properties of these resolvable designs restrict the search space. A consistency checking based on the principle of double counting and the orbit-stabilizer theorem is utilized to verify the computation results.