Two New $1$-rotational $(36,9,8)$ and $(40,10,9)$ RBIBDs

Abstract

We formulate the construction of 1-rotational difference families as a combinatorial optimization problem. A tabu search algorithm is used to find an optimal solution to the optimization problem for various 1-rotational difference family parameters. In particular, we construct two new 1-rotational difference families which lead to an equal number of new 1-rotational RBIBDs with parameters: $(36,9,8)$ and $(40,10,9)$. Our algorithm also was able to construct six non-isomorphic $(36,9,8)$ and three $(40,10,9)$ RBIBDs