  Classifying orthogonal arrays is a well known important class of problems
that asks for finding all non-isomorphic, non-negative integer solutions to a
class of systems of constraints. Solved instances are scarce. We develop two
new methods based on finding all non-isomorphic solutions of two novel integer
linear programming formulations for classifying all non-isomorphic OA(N,k,s,t)
given a set of all non-isomorphic OA(N,k-1,s,t). We also establish the concept
of orthogonal design equivalence of OA(N,k,2,t) to reduce the number of integer
linear programs (ILPs) whose all non-isomorphic solutions need to be enumerated
by our methods. For each ILP, we determine the largest group of permutations
that can be exploited with the branch-and-bound (B&B) with isomorphism pruning
algorithm of Margot [Discrete Optim~4 (2007), 40-62] without losing isomorphism
classes of OA(N,k,2,t). Our contributions brought the classifications of all
non-isomorphic OA(160,k,2,4) for k=9,10 and OA(176,k,2,4) for k=5,6,7,8,9,10
within computational reach. These are the smallest s=2, t=4 cases for which
classification results are not available in the literature.
