Constructions Of Asymmetric Orthogonal Arrays Of Strength Three

Orthogonal arrays are very important in all areas of human investigation,the definition is simple,and there are many constructions at the present.But constructing particular orthogonal arrays are still topic worthy of further research for practical applications.This paper uses columnar Latin squares and replacement method to construct a family of asymmetric orthogonal arrays of strength three.Nguyen?2008?used Latin squares and started from complementary pairs of grids to construct asymmetric orthogonal arrays OA?96,6 × 4²×2^k,3??1 ? k ? 5?of strength three,where the arrays with run sizes 96.The second chapter of this paper proposes a concept of the complementary column pairs,starts from the complementary column pairs to construct asymmetric orthogonal arrays OA?96,6 × 4²× 2^k,3?,k ? 3.The results of the associated theoretical are more general.On the basis of the arrays,this paper further find the existence of a symmetric orthogonal array OA?96,6 × 4²× 2⁵,3?.Suen et al.?2001?proposed a replacement method for factor levels s,where s is a power of two.That is to say,they started from a symmetric orthogonal array,through a series of replacement procedures to construct a tight asymmetric orthogonal array of strength three,where the array with run sizes 2s3.In the third chapter of this paper,we further explore the replacement method to general situations,that is to say,we start from orthogonal arrays OA(s^m,n,?s^r?× s^n-1,3)?s = pk,p is a prime,k is an integer?,through a series of replacement procedures to construct asymmetric orthogonal arrays OA(ps^m,n,?ps^r?×s^n-1,3),then replacing a factor with pk-level by several p-level factors to construct a family of asymmetric orthogonal arrays OA?ps^m,t + u + 1,?psr?× s^t× p^u,3?,t,u are integers.This replacement method without disturbing the orthogonality of the arrays.This paper mainly aims at the conditions of p = 2,3,a new family of asymmetric orthogonal arrays are obtained,some of the obtained are tight.Finally,this paper also discusses the problem of the number of replacing a 3k-level factor by 3-level factors.