Isomorph-rejection: Theory and an application

We describe, in some detail, classification algorithms that are suitable for several common searching tasks in combinatorics. We first consider the theory of two orderly generation algorithms. The first is due to Faradzev and Read (independently), while the second is due to McKay. After that, we use the later one in the classification of {0,1} matrices with v rows and b columns (where v, b ≤ 12) so that each row and each column has constant row sum and constant column sum, respectively. As a result, there are 317,579,783 non-isomorphic 12 x 12 matrices over {0,1} with rows and columns summing to 4.