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Combinatorial covering arrays have been used in several testing approaches. This paper first discusses some existing methods for finding such arrays. Then a SAT-based approach and a backtracking search algorithm are presented to solve the problem. A novel pruning strategy called SCEH is proposed to increase the efficiency of the methods. Several existing search heuristics and symmetry breaking techniques are also used in the backtracking search algorithm. Lastly, this paper introduces a tool called EXACT (EXhaustive seArch of Combinatorial Test suites) which implements all the above techniques to construct the covering arrays automatically. The experimental results show that our backtracking search method outperforms other methods in many small size cases.