Covering arrays of higher strength from permutation vectors

A covering array CA(N;t,k,v) is an N × k array such that every N × t sub‐array contains all t‐tuples from v symbols at least once, where t is the strength of the array. Covering arrays are used to generate software test suites to cover all t‐sets of component interactions. We introduce a combinatorial technique for their construction, focussing on covering arrays of strength 3 and 4. With a computer search, covering arrays with improved parameters have been found. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 202–213, 2006     

A construction of maximal linearly independent array

Let M be a subset of r-dimensional vector space Vτ （F2） over a finite field F2, consisting of n nonzero vectors, such that every t vectors of M are linearly independent over F2. Then M is called （n, t）-linearly independent array of length n over Vτ（F2）. The （n, t）-linearly independent array M that has the maximal number of elements is called the maximal （r, t）-linearly independent array, and the maximal number is denoted by M（r, t）. It is an interesting combinatorial structure, which has many applications in cryptography and coding theory. It can be used to construct orthogonal arrays, strong partial balanced designs. It can also be used to design good linear codes, In this paper, we construct a class of maximal （r, t）-linearly independent arrays of length r ＋ 2, and provide some enumerator theorems.     

Further results on the existence of nested orthogonal arrays

Nested orthogonal arrays provide an option for designing an experimental setup consisting of two experiments, the expensive one of higher accuracy being nested in a larger and relatively less expensive one of lower accuracy. We denote by OA_(λ, μ)(t, k, (v, w)) (or OA(t, k, (v, w)) if λ = μ = 1) a (symmetric) orthogonal array OA _λ (t, k, v) with a nested OA _μ (t, k, w) (as a subarray). It is proved in this article that an OA(t, t + 1,(v, w)) exists if and only if v ≥ 2w for any positive integers v, w and any strength t ≥ 2. Some constructions of OA_(λ, μ)(t, k, (v, w))′s with λ ≠ μ and k ? t > 1 are also presented.     

Linear hash families and forbidden configurations

The classical orthogonal arrays over the finite field underlie a powerful construction of perfect hash families. By forbidding certain sets of configurations from arising in these orthogonal arrays, this construction yields previously unknown perfect, separating, and distributing hash families. When the strength s of the orthogonal array, the strength t of the hash family, and the number of its rows are all specified, the forbidden sets of configurations can be determined explicitly. Each forbidden set leads to a set of equations that must simultaneously hold. Hence computational techniques can be used to determine sufficient conditions for a perfect, separating, and distributing hash family to exist. In this paper the forbidden configurations, resulting equations, and existence results are determined when (s, t) ∈ {(2, 5), (2, 6), (3, 4), (4, 3)}. Applications to the existence of covering arrays of strength at most six are presented.      

The Existence of a Symmetric Transversal Design STD 7[21; 3]

In this note we construct a symmetric transversal design STD₇[21; 3]. Therefore there exists an orthogonal array OA(63, 21, 3, 2). Communicated by: J.D. Key     

Products of mixed covering arrays of strength two

A covering array CA(N;t,k, v is an N × k array such that every N × t subarray contains all t‐tuples from v symbols at least once, where t is the strength of the array. Covering arrays are used to generate software test suites to cover all t‐sets of component interactions. The particular case when t = 2 (pairwise coverage) has been extensively studied, both to develop combinatorial constructions and to provide effective algorithmic search techniques. In this paper, a simple “cut‐and‐paste” construction is extended to covering arrays in which different columns (factors) admit different numbers of symbols (values); in the process an improved recursive construction for covering arrays with t = 2 is derived. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 124–138, 2006     

Ideal ramp schemes and augmented orthogonal arrays

Ramp schemes were invented in 1985 by C.R. Blakley and C. Meadows. An $(s,t,n)$-ramp scheme is a generalization of a threshold scheme in which there are two thresholds. Recently, D.R. Stinson established the equivalence of ideal ramp schemes and augmented orthogonal arrays. In this study, some new construction methods for augmented orthogonal arrays are presented and then some new augmented orthogonal arrays are obtained; furthermore, we also provide parameter situations where ideal ramp schemes exist for these obtained augmented orthogonal arrays.     

Roux-type constructions for covering arrays of strengths three and four

A covering array CA(N;t,k,v) is an N × k array such that every N × t sub-array contains all t-tuples from v symbols at least once, where t is the strength of the array. Covering arrays are used to generate software test suites to cover all t-sets of component interactions. Recursive constructions for covering arrays of strengths 3 and 4 are developed, generalizing many “Roux-type” constructions. A numerical comparison with current construction techniques is given through existence tables for covering arrays.      

On schematic orthogonal arraysOA(q t,k, q,t)

Am × k matrixA, with entries from a set ofq 2 elements, is called an orthogonal arrayOA(m, k, q, t) (t 2) if eachm × t submatrix ofA contains all possible 1 ×t row vectors with the same frequency(m = q ^t ). We call the array schematic if the set of rows ofA forms an association scheme with the relations determined by the Hamming distance. In this paper we determine the schematic orthogonal arraysOA(q ^t ,k, q, t) with2t – 1 > k.     

Bounds on orthogonal arrays and resilient functions

The only known general bounds on the parameters of orthogonal arrays are those given by Rao in 1947 [J. Roy. Statist. Soc. 9 (1947), 128–139] for general OA_γ(t,k,v) and by Bush [Ann. Math. Stat. 23, (1952), 426–434] [3] in 1952 for the special case γ = 1. We present an algebraic method based on characters of homocyclic groups which yields the Rao bounds, the Bush bound in case t ? v, and more importantly a new explicit bound which for large values of t (the strength of the array) is much better than the Rao bound. In the case of binary orthogonal arrays where all rows are distinct this bound was previously proved by Friedman [Proc. 33rd IEEE Symp. on Foundations of Comput. Sci., (1992), 314–319] in a different setting. We also note an application to resilient functions. © 1995 John Wiley & Sons, inc.     

Covering Arrays of Strength Three

A covering array of size N, degree k, order v and strength t is a k × N array with entries from a set of v symbols such that in any t × N subarray every t × 1 column occurs at least once. Covering arrays have been studied for their applications to drug screening and software testing. We present explicit constructions and give constructive upper bounds for the size of a covering array of strength three.     

On the existence of nested orthogonal arrays

A nested orthogonal array is an OA(N,k,s,g) which contains an OA(M,k,r,g) as a subarray. Here r<s and M<N. Necessary conditions for the existence of such arrays are obtained in the form of upper bounds on k, given N, M, s, r and g. Examples are given to show that these bounds are quite powerful in proving nonexistence. The link with incomplete orthogonal arrays is also indicated.     

Group construction of covering arrays

A covering array t‐CA (n, k, g) is a k × n array on a set of g symbols with the property that in each t × n subarray, every t × 1 column appears at least once. This paper improves many of the best known upper bounds on n for covering arrays, 2‐CA (n, k, g) with g + 1 ≤ k ≤ 2g, for g = 3 · · · 12 by a construction which in many of these cases produces a 2‐CA (n, k, g) with n = k (g ? 1) + 1. The construction is an extension of an algebraic method used by Chateauneuf, Colbourn, and Kreher which uses an array and a group action on the array. © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 70–77, 2005.     

On linear programming bounds for nets

It is well known that there are close connections between low-discrepancy point sets and sequences on the one hand, and certain combinatorial and algebraic structures on the other hand. E. g., Niederreiter [1] showed the equivalence between (t, t + 2, s)-nets and orthogonal arrays of strength 2. Some years later this was generalized and made precise in the work of Lawrence [2] as well as Mullen and Schmid [3] by introducing ordered orthogonal arrays. This large class of combinatorial structures yields both new constructions and bounds for the existence of nets and sequences. The linear programming bound for ordered orthogonal arrays was first derived by Martin and Stinson [4]. As in the case of error-correcting codes and orthogonal arrays, it yields a very strong bound for ordered orthogonal arrays, and consequently for nets and sequences. Solving linear programming problems in exact arithmetics is very time-consuming. Using different approaches to reduce the computing time, we have calculated the linear programming bound for a wide parameter range. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constructions of nets via OAs of strength 3 with a prescribed property

The idea of (t, m, s)‐nets was proposed by Niederreiter in 1987. Such nets are highly uniform point distributions in s‐dimensional unit cubes and useful in numerical analysis. It is by now well known that (t, m, s)‐nets can be equivalently described in terms of ordered orthogonal arrays (OOAs). In this article, we describe an equivalence between an OOA and an orthogonal array (OA) with all its derived orthogonal subarrays being resolvable. We then present a number of constructions for OAs where all their derived orthogonal subarrays are resolvable. These results are finally combined to give new series of (t, m, s)‐nets. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:144‐155, 2011     

The spectrum of 2‐idempotent 3‐quasigroups with conjugate invariant subgroups

A ternary quasigroup (or 3‐quasigroup) is a pair (N, q) where N is an n‐set and q(x, y, z) is a ternary operation on N with unique solvability. A 3‐quasigroup is called 2‐idempotent if it satisfies the generalized idempotent law: q(x, x, y) = q(x, y, x) = q(y, x, x)=y. A conjugation of a 3‐quasigroup, considered as an OA(3, 4, n), $({{N}},{\mathcal{B}})$, is a permutation of the coordinate positions applied to the 4‐tuples of ${\mathcal{B}}$. The subgroup of conjugations under which $({{N}},{\mathcal{B}})$ is invariant is called the conjugate invariant subgroup of $({{N}},{\mathcal{B}})$. In this article, we determined the existence of 2‐idempotent 3‐quasigroups of order n, n≡7 or 11 (mod 12) and n≥11, with conjugate invariant subgroup consisting of a single cycle of length three. This result completely determined the spectrum of 2‐idempotent 3‐quasigroups with conjugate invariant subgroups. As a corollary, we proved that an overlarge set of Mendelsohn triple system of order n exists if and only if n≡0, 1 (mod 3) and n≠6. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 292–304, 2010     

Four Mutually Orthogonal Latin Squares of Order 14

The paper gives example of orthogonal array OA(6, 14) obtained from a difference matrix . The construction is equivalent to four mutually orthogonal Latin squares (MOLS) of order 14. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 363–367, 2012     

Complete enumeration of pure‐level and mixed‐level orthogonal arrays

We specify an algorithm to enumerate a minimum complete set of combinatorially non‐isomorphic orthogonal arrays of given strength t, run‐size N, and level‐numbers of the factors. The algorithm is the first one handling general mixed‐level and pure‐level cases. Using an implementation in C, we generate most non‐trivial series for t=2, N≤28, t=3, N≤64, and t=4, N≤168. The exceptions define limiting run‐sizes for which the algorithm returns complete sets in a reasonable amount of time. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 123–140, 2010     

The equivalence between optimal detecting arrays and super-simple OAs

The notion of a detecting array (DTA) was proposed, recently, by Colbourn and McClary in their research on software interaction tests. Roughly speaking, testing with a (d, t)−DTA(N, k, v) can locate d interaction faults and detect whether there are more than d interaction faults. In this paper, we establish a general lower bound on sizes of DTAs and explore an equivalence between optimal DTAs and super-simple orthogonal arrays (OAs). Taking advantage of this equivalence, a great number of DTAs are constructed, which are all optimal in the sense of their sizes. In particular, an optimal (2, t)−DTA(N, 5, v) of strength t = 2 or 3 is shown to exist whenever v ≥ 3 excepting (t, v) ? {(2, 3), (2, 6),(3, 4), (3, 6)}{(t, v) \in \{(2, 3), (2, 6),(3, 4), (3, 6)\}} .     

A family of pandiagonal bimagic squares based on orthogonal arrays

In this article we give a construction of pandiagonal bimagic squares by means of four‐dimensional bimagic rectangles, which can be obtained from orthogonal arrays with special properties. In particular, we show that there exists a normal pandiagonal bimagic square of order n⁴ for all positive integer n≥7 such that gcd(n, 30) = 1 , which gives an answer to problem 22 of Abe in [Discrete Math 127 (1994), 3–13]. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:427‐438, 2011