Constructions of new orthogonal arrays and covering arrays of strength three

A covering array of size N, strength t, degree k, and order v, or a CA ( N ; t , k , v ) in short, is a k × N array on v symbols. In every t × N subarray, each t-tuple column vector occurs at least once. When ‘at least’ is replaced by ‘exactly’, this defines an orthogonal array, OA ( t , k , v ) . A difference covering array, or a DCA ( k , n ; v ) , over an abelian group G of order v is a k × n array ( a i j ) ( 1 ⩽ i ⩽ k , 1 ⩽ j ⩽ n ) with entries from G, such that, for any two distinct rows l and h of D ( 1 ⩽ l < h ⩽ k ), the difference list Δ l h = { d h 1 − d l 1 , d h 2 − d l 2 , … , d h n − d l n } contains every element of G at least once. ng arrays have important applications in statistics and computer science, as well as in drug screening. In this paper, we present two constructive methods to obtain orthogonal arrays and covering arrays of strength 3 by using DCAs. As a consequence, it is proved that there are an OA ( 3 , 5 , v ) for any integer v ⩾ 4 and v ≢ 2 (mod 4), and an OA ( 3 , 6 , v ) for any positive integer v satisfying gcd ( v , 4 ) ≠ 2 and gcd ( v , 18 ) ≠ 3 . It is also proved that the size CAN ( 3 , k , v ) of a CA ( N ; 3 , k , v ) cannot exceed v 3 + v 2 when k = 5 and v ≡ 2 (mod 4), or k = 6 , v ≡ 2 (mod 4) and gcd ( v , 18 ) ≠ 3 .