On the construction and existence of orthogonal arrays with three levels and indexes 1 and 2. (English) Zbl 0886.05033

Hedayat, S.; Stufken, John; Su, Guoqin

Ann. Stat. 25, No. 5, 2044-2053 (1997).

An \(N\times k\) matrix \(A\) with entries from a set \(S\) of cardinality \(s\) is called an orthogonal array of strength \(t\), \(1\leq t\leq k\), denoted by \(\text{OA}(N,k,s,t)\), if every \(N\times t\) submatrix of \(A\) contains each \(t\)-tuple based on \(S\) equally often as a row. The common frequency \(\lambda= N/2^t\) with which each of the \(t\)-tuples appears as a row in a submatrix is called the index of the array. The authors provide the construction of orthogonal arrays with three levels and index 2. For strengths greater than or equal to 2, they show that orthogonal arrays with 3 levels and index 1 are unique. Finally they establish that the maximum number of factors is \(k\) for orthogonal arrays with 3 levels and index 2.
The paper disappoints and mentions very little, if any, on connections of OA’s with codes, linear codes and recent results on these topics (e.g., B. L. Raktoe and H. Pesotan [Math. Methods Stat. 5, No. 2, 244-252 (1996)] as related to saturated \(s^{k-p}_R\) fractional factorial designs, where \(N= s^q\), \(q=k-p\), and \(R= t+1\), \(s\) is a prime or prime power).

Reviewer: B.L.Raktoe (Singapore)

MSC:

05B15	Orthogonal arrays, Latin squares, Room squares
62K15	Factorial statistical designs
94B05	Linear codes (general theory)

Keywords:

orthogonal array; linear codes; fractional factorial designs