Difference schemes via computerized searches

Some difference schemes cannot be constructed by any known algebraic method. We developed a sequential-search algorithm that makes larger difference schemes by appending columns to smaller difference schemes. We used it to find the known, but obscure difference scheme D(15,9,3) and the apparently new difference schemes D(21,12,3), D(20,10,4), D(24,18,4), and D(15,8,5), all of which are useful in constructing orthogonal arrays.