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Component based software development is prone to unexpected interaction faults. The goal is to test as many-potential interactions as is feasible within time and budget constraints. Two combinatorial objects, the orthogonal array and the covering array, can be used to generate test suites that provide a guarantee for coverage of all t-sets of component interactions in the case when the testing of all interactions is not possible. Methods for construction of these types of test suites have focused on two main areas. The first is finding new algebraic constructions that produce smaller test suites. The second is refining computational search algorithms to find smaller test suites more quickly. In this paper we explore one method for constructing covering arrays of strength three that combines algebraic constructions with computational search. This method leverages the computational efficiency and optimality of size obtained through algebraic constructions while benefiting from the generality of a heuristic search. We present a few examples of specific constructions and provide some new bounds for some strength three covering arrays.