Construction of orthogonal arrays with repetitions of two distinct rows

Orthogonal arrays with repeated rows are applied in various aspects such as statistical experimental design and pure error estimation, quantum information and quantum communication, and scientific computing and computer theory. However, existing research only focuses on an orthogonal array in which a row is repeated, while research on an orthogonal array with repetitions of two distinct rows remains scarce. In this paper, we mainly focus on such an array. As well known, a bound is quite powerful in proving nonexistence. Because of the complexity of its structures, from two aspects we derive a bound for ones of strength 2. Based on this bound, we define optimal, x -optimal and ( x , m 1 , m 2 ) -optimal arrays. We further present a general construction method via difference schemes and obtain a large number of such arrays. Moreover, for the cases of high strength, two efficient construction methods are proposed to produce a large number of new arrays. Some selective arrays are tabulated for practical uses.