The Existence of Mixed Orthogonal Arrays with Four and Five Factors of Strength Two

Abstract

Symmetric orthogonal arrays and mixed orthogonal arrays are useful in the design of various experiments. They are also a fundamental tool in the construction of various combinatorial configurations. In this paper, we investigated the mixed orthogonal arrays with four and five factors of strength two, and proved that the necessary conditions of such mixed orthogonal arrays are also sufficient with several exceptions and one possible exception. C 2013 Wiley Periodicals, Inc. J. Combin. Designs 22: 323-342, 2014

Construction of Asymmetric Orthogonal Arrays of Strength Three via a Replacement Method

“…Hence an array OA(2s 5 , s 3 − (s − 2)t, (2s 2 ) × s t × 2 (s 2 +s+1−t)(s−1) , 3) can be constructed for 0 ≤ t ≤ s 2 + s + 1. The tightness of the array follows from (1).…”

“…More recent work on orthogonal arrays of strength two include those by Suen and Kuhfeld and Chen et al. . Methods of constructing asymmetric orthogonal arrays of strength greater than two have not been studied as extensively as those of strength two.…”

“…The construction of asymmetric orthogonal arrays of strength two have been studied extensively, and one may refer to Hedayat et al [3] for a comprehensive account of these. More recent work on orthogonal arrays of strength two include those by Suen and Kuhfeld [9] and Chen et al [1]. Methods of constructing asymmetric orthogonal arrays of strength greater than two have not been studied as extensively as those of strength two.…”

“…An orthogonal array is called tight if it has the smallest possible number of rows (given the other parameters of the array). Thus, an orthogonal array OA(N, n, m 1 × · · · × m n , 3) of strength three will be called tight if the number of rows, N, attains the lower bound given by (1).…”

Quantum k-uniform states for heterogeneous systems from irredundant mixed orthogonal arrays

“…Based on Hamming distances of orthogonal arrays (OAs) with difference schemes and orthogonal partitions, Pang et al [22] explicitly constructed infinite classes of k-uniform states for k = 2, 3. Furthermore, by using the product construction [25], Bush's construction, binary double-error-correcting BCH codes and expansive replacement method [26], Pang et al [27] constructed infinitely classes of k-uniform states for k ≥ 4. In addition, k-uniform states can be constructed from mutually orthogonal Latin squares and Latin cubes [21], graph states [28] and quantum combinatorial designs [1].…”