Constructions Of Mixed Orthogonal Arrays For T = 3, K = 4, 5

Orthogonal array is a kind of combinatorial structure to study experimental designs.which was introduced by the statistician C. R. Rao in 1947. In the field of combination design and experimental design, orthogonal array occupies an important position. The application of orthogonal arrays is extremely widespread, which has prompted many scholars to devote to the research of orthogonal arrays and to obtain many important results. With the development of science and technology, several experimental factors of different levels are required in the actual experiment. Therefore, C.R.Rao put forward the concept of mixed orthogonal array in 1973.Many constructions and results of orthogonal array and many references are listed in Orthogonal Arrays: Theory and Applications, which provides the theoretical basis for study of orthogonal array. So far, the construction methods and results about mixed orthogonal array of strength t ≥ 3 is very few, which has restricted the application of orthogonal array on experimental designs. In this paper, we mainly investigate the mixed orthogonal array with four and five factors.In this paper, we discussed different cases according to the relationship between different factor levels. We completely solved the existence of mixed orthogonal arrays with four factors of strength three, mainly solved the existence of mixed orthogonal arrays with five factors of strength three(MOA(N; abcde, 3), a, b, c, d, e is positive integer,N = l.c.m.{uvk : {u, v, k} (?) {a, b, c, d, e}}), possibly except two types:(1){abcde : x ∈{a, b, c, d, e}, x ≡ 2(mod 4)}, where a, b, c, d, e can take the same number;(2){abcde :x ∈ {a, b, c, d}, x ≡ 2(mod 4), e ≡ 0(mod 4)}, where a, b, c, d can take the same number.