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     

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.      

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 state of strength‐three covering arrays

A covering array of size N, strength t, degree k, and order υ is a k × N array on υ symbols in which every t × N subarray contains every possible t × 1 column at least once. We present explicit constructions, constructive upper bounds on the size of various covering arrays, and compare our results with those of a commercial product. Applications of covering arrays include software testing, drug screening, and data compression. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 217–238, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10002     

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.     

Constructions of covering arrays of strength five

A covering array of size N, strength t, degree k and order v, or a CA(N; t, k, v) in short, is an N × k array on v symbols. In every N × t subarray, each t-tuple occurs in at least one row. Covering arrays have been studied for their significant applications to generating software test suites to cover all t-sets of component interactions. In this paper, we present two constructive methods to obtain covering arrays of strength 5 by using difference covering arrays and holey difference matrices with a prescribed property. As a consequence, some new upper bounds on the covering numbers are derived.     

The decomposability of simple orthogonal arrays on 3 symbols having t + 1 rows and strength t

It is well‐known that all orthogonal arrays of the form OA(N, t + 1, 2, t) are decomposable into λ orthogonal arrays of strength t and index 1. While the same is not generally true when s = 3, we will show that all simple orthogonal arrays of the form OA(N, t + 1, 3, t) are also decomposable into orthogonal arrays of strength t and index 1. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 442–458, 2000     

Constructing strength three covering arrays with augmented annealing

A covering arrayCA(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. One application of these objects is to generate software test suites to cover all t-sets of component interactions. Methods for construction of covering arrays for software testing have focused on two main areas. The first is finding new algebraic and combinatorial constructions that produce smaller covering arrays. The second is refining computational search algorithms to find smaller covering arrays more quickly. In this paper, we examine some new cut-and-paste techniques for strength three covering arrays that combine recursive combinatorial constructions with computational search; when simulated annealing is the base method, this is augmented annealing. This method leverages the computational efficiency and optimality of size obtained through combinatorial constructions while benefiting from the generality of a heuristic search. We present a few examples of specific constructions and provide new bounds for some strength three covering arrays.     

Covering arrays with mixed alphabet sizes

Covering arrays with mixed alphabet sizes, or simply mixed covering arrays, are natural generalizations of covering arrays that are motivated by applications in software and network testing. A (mixed) covering array A of type is a k × N array with the cells of row i filled with elements from ? and having the property that for every two rows i and j and every ordered pair of elements (e,f) ∈ ? × ?, there exists at least one column c, 1 ≤ c ≤ N, such that A_i,c = e and A_j,c = f. The (mixed) covering array number, denoted by , is the minimum N for which a covering array of type with N columns exists. In this paper, several constructions for mixed covering arrays are presented, and the mixed covering array numbers are determined for nearly all cases with k = 4 and for a number of cases with k = 5. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 413–432, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10059     

Covering arrays of strength 3 and 4 from holey difference matrices

A covering array CA(N; t, k, v) is an N × k array with entries from a set X of v symbols such that every N × t sub-array contains all t-tuples over X at least once, where t is the strength of the array. The minimum size N for which a CA(N; t, k, v) exists is called the covering array number and denoted by CAN(t, k, v). Covering arrays are used in experiments to screen for interactions among t-subsets of k components. One of the main problems on covering arrays is to construct a CA(N; t, k, v) for given parameters (t, k, v) so that N is as small as possible. In this paper, we present some constructions of covering arrays of strengths 3 and 4 via holey difference matrices with prescribed properties. As a consequence, some of known bounds on covering array number are improved. In particular, it is proved that (1) CAN(3, 5, 2v) ≤ 2v ²(4v + 1) for any odd positive integer v with gcd(v, 9) ≠ 3; (2) CAN(3, 6, 6p) ≤ 216p ³ + 42p ² for any prime p > 5; and (3) CAN(4, 6, 2p) ≤ 16p ⁴ + 5p ³ for any prime p ≡ 1 (mod 4) greater than 5.     

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.      

On the structure of small strength‐2 covering arrays

A covering array $\text{CA}\left(N;t,k,v\right)$ of strength $t$ is an $N\times k$ array of symbols from an alphabet of size $v$ such that in every $N\times t$ subarray, every $t$‐tuple occurs in at least one row. A covering array is optimal if it has the smallest possible $N$ for given $t$, $k$, and $v$, and uniform if every symbol occurs $?N∕v?$ or $?N∕v?$ times in every column. Before this paper, the only known optimal covering arrays for $t=2$ were orthogonal arrays, covering arrays with $v=2$ constructed from Sperner's Theorem and the Erd?s‐Ko‐Rado Theorem, and 11 other parameter sets with $v>2$ and $N>v^2$. In all these cases, there is a uniform covering array with the optimal size. It has been conjectured that there exists a uniform covering array of optimal size for all parameters. In this paper, a new lower bound as well as structural constraints for small uniform strength‐2 covering arrays is given. Moreover, covering arrays with small parameters are studied computationally. The size of an optimal strength‐2 covering array with $v>2$ and $N>v^2$ is now known for 21 parameter sets. Our constructive results continue to support the conjecture.     

Mixed covering arrays on graphs

Covering arrays have applications in software, network and circuit testing. In this article, we consider a generalization of covering arrays that allows mixed alphabet sizes as well as a graph structure that specifies the pairwise interactions that need to be tested. Let k and n be positive integers, and let G be a graph with k vertices v₁,v₂,…, v_k with respective vertex weights g₁ ≤ g₂ ≤ … ≤ g_k. A mixed covering array on G, denoted by , is an n × k array such that column i corresponds to v_i, cells in column i are filled with elements from ?_gi and every pair of columns i,j corresponding to an edge v_i,v_j in G has every possible pair from ?_gi × ?_gj appearing in some row. The number of rows in such array is called its size. Given a weighted graph G, a mixed covering array on G with minimum size is called optimal. In this article, we give upper and lower bounds on the size of mixed covering arrays on graphs based on graph homomorphisms. We provide constructions for covering arrays on graphs based on basic graph operations. In particular, we construct optimal mixed covering arrays on trees, cycles and bipartite graphs; the constructed optimal objects have the additional property of being nearly point balanced. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 393–404, 2007     

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     

There are no Barker arrays having more than two dimensions

Davis, Jedwab and Smith recently proved that there are no 2-dimensional Barker arrays except of size 2 × 2. We show that the existence of a (d + 1)-dimensional Barker array implies the existence of a d-dimensional Barker array with the same number of ± 1 elements. We deduce that there are no Barker arrays having more than two dimensions, as conjectured by Dymond in 1992.      

Generalized covering designs and clique coverings

Inspired by the “generalized t‐designs” defined by Cameron [P. J. Cameron, Discrete Math 309 (2009), 4835–4842], we define a new class of combinatorial designs which simultaneously provide a generalization of both covering designs and covering arrays. We then obtain a number of bounds on the minimum sizes of these designs, and describe some methods of constructing them, which in some cases we prove are optimal. Many of our results are obtained from an interpretation of these designs in terms of clique coverings of graphs. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:378‐406, 2011     

Covering arrays from cyclotomy

For a prime power ${q \equiv 1 (mod{v})}$ , the q × q cyclotomic matrix, whose entries are the discrete logarithms modulo v of the entries in the addition table of ${\mathbb{F}_q}$ , has been shown using character theoretic arguments to produce an ${\varepsilon}$ -biased array, provided that q is large enough as a function of v and ${\varepsilon}$ . A suitable choice of ${\varepsilon}$ ensures that the array is a covering array of strength t when ${q > t^2 v^{4t}}$ . On the other hand, when v = 2, using a different character-theoretic argument the matrix has been shown to be a covering array of strength t when ${q > t^2 2^{2t-2}}$ . The restrictions on ${\varepsilon}$ -biased arrays are more severe than on covering arrays. This is exploited to prove that for all v ≥ 2, the matrix is a covering array of strength t whenever ${q > t^2 v^{2t}}$ , again using character theory. A number of constructions of covering arrays arise by developing and extending the cyclotomic matrix. For each construction, extensive computations for various choices of t and v are reported that determine the precise set of small primes for which the construction produces a covering array. As a consequence, many covering arrays are found when q is smaller than the bound ${t^2 v^{2t}}$ , and consequences for the existence of covering arrays reported.     

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.     

Binary consecutive covering arrays

A k × n array with entries from a q-letter alphabet is called a t-covering array if each t × n submatrix contains amongst its columns each one of the q ^t different words of length t that can be produced by the q letters. In the present article we use a probabilistic approach based on an appropriate Markov chain embedding technique, to study a t-covering problem where, instead of looking at all possible t × n submatrices, we consider only submatrices of dimension t × n with its rows being consecutive rows of the original k × n array. Moreover, an exact formula is established for the probability distribution function of the random variable, which enumerates the number of deficient submatrices (i.e., submatrices with at least one missing word, amongst their columns), in the case of a k × n binary matrix (q = 2) obtained by realizing kn Bernoulli variables.     

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