Incomplete orthogonal arrays and idempotent orthogonal arrays

First, we shall define idempotent orthogonal arrays and notice that idempotent orthogonal array of strength two are idempotent mutually orthogonal quasi-groups. Then, we shall state some properties of idempotent orthogonal arrays. Next, we shall prove that, starting from an incomplete orthogonal array T-E\F based on E and F subset of E, from an orthogonal array T-G based on G = E - F and from an idempotent orthogonal array T-H based on H, we are able to construct an incomplete orthogonal array T-(F boolean OR(GxH)\F) based on F boolean OR (G x H) and F. Finally, we shall show the relationship between the construction which lead us to this result and the singular direct product of mutually orthogonal quasi-groups given by Sade [5].