Construction of orthogonal arrays of strength three by augmented difference schemes

Difference schemes are a powerful tool for developing orthogonal arrays (OAs). In this study, we define an augmented difference scheme, i.e., a generalization of a difference scheme, and present a general method for constructing such schemes of strength three. As an application of the proposed method, we construct difference schemes with any number of levels, including D3(d2, c, d) with the maximal value c for any prime power d. Furthermore, using the newly constructed difference schemes, a large number of new OAs of strength three can be infinitely obtained, including many tight arrays. Accordingly, we provide a positive answer to two open problems: how to develop new methods for the construction of difference schemes of high strength with the ultimate goal of obtaining better OAs of high strength, and how to develop better methods and tools for the construction of OAs.