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.     

Covering arrays of strength three from extended permutation vectors

A covering array \(\text{ CA }(N;t,k,v)\) is an \(N\times k\) array such that in every \(N\times t\) subarray each possible t-tuple over a v-set appears as a row of the subarray at least once. The integers t and v are respectively the strength and the order of the covering array. Let v be a prime power and let \({\mathbb {F}}_v\) denote the finite field with v elements. In this work the original concept of permutation vectors generated by a \((t-1)\)-tuple over \({\mathbb {F}}_v\) is extended to vectors generated by a t-tuple over \({\mathbb {F}}_v\). We call these last vectors extended permutation vectors (EPVs). For every prime power v, a covering perfect hash family \(\text{ CPHF }(2;v^2-v+3,v^3,3)\) is constructed from EPVs given by subintervals of a linear feedback shift register sequence over \({\mathbb {F}}_v\). When \(v\in \{7,9,11,13,16,17,19,23,25\}\) the covering array \(\text{ CA }(2v^3-v;3,v^2-v+3,v)\) generated by \(\text{ CPHF }(2;v^2-v+3,v^3,3)\) has less rows than the best-known covering array with strength three, \(v^2-v+3\) columns, and order v. CPHFs formed by EPVs are also constructed using simulated annealing; in this case the results improve the size of eighteen covering arrays of strength three.     

An extension of a construction of covering arrays

By Raaphorst et al, for a prime power $q$, covering arrays (CAs) with strength 3 and index 1, defined over the alphabet ${\mathbb{F}}_q$, were constructed using the output of linear feedback shift registers defined by cubic primitive polynomials in ${\mathbb{F}}_q\left[x\right]$. These arrays have $2q^3?1$ rows and $q^2+q+1$ columns. We generalize this construction to apply to all polynomials; provide a new proof that CAs are indeed produced; and analyze the parameters of the generated arrays. Besides arrays that match the parameters of those of Raaphorst et al, we obtain arrays matching some constructions that use Chateauneuf‐Kreher doubling; in both cases these are some of the best arrays currently known for certain parameters.     

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     

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     

On the existence of retransmission permutation arrays

We investigate retransmission permutation arrays (RPAs) that are motivated by applications in overlapping channel transmissions. An RPA is an $n\times n$ array in which each row is a permutation of $\{1,\dots ,n\}$, and for $1?i?n$, all $n$ symbols occur in each $i\times ?\frac{n}{i}?$ rectangle in specified corners of the array. The array has types 1, 2, 3 and 4 if the stated property holds in the top left, top right, bottom left and bottom right corners, respectively. It is called latin if it is a latin square. We show that for all positive integers $n$, there exists a type-1, 2, 3, 4 $\mathrm{RPA}\left(n\right)$ and a type-1, 2 latin $\mathrm{RPA}\left(n\right)$.     

Variable strength covering arrays

We define and study variable strength covering arrays (also called covering arrays on hypergraphs), which are generalizations of covering arrays and covering arrays on graphs. Variable strength covering arrays have the potential for use in software testing, allowing the engineer to omit the parameter combinations known to not interact in order to reduce the number of tests required. The present paper shows that variable strength covering arrays are relevant combinatorial objects that have deep connections with hypergraph homomorphisms and generalize other important combinatorial designs. We give optimal constructions for special types of hypergraphs, constructions based on columns with uniform occurrence of symbols, and constructions for mixed alphabets.     

Balanced arrays of strength two from block designs

Some constructions of balanced arrays of strength two are provided by use of rectangular designs, group divisible designs, and nested balanced incomplete block designs. Some series of such arrays are also presented as well as orthogonal arrays, with illustrations. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 303–312, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10016     

Covering arrays for some equivalence classes of words

Covering arrays for words of length over a ‐letter alphabet are arrays with entries from the alphabet so that for each choice of columns, each of the ‐letter words appears at least once among the rows of the selected columns. We study two schemes in which all words are not considered to be different. In the first case known as partitioning hash families, words are equivalent if they induce the same partition of a element set. In the second case, words of the same weight are equivalent. In both cases, we produce logarithmic upper bounds on the minimum size of a covering array. Definitive results for , as well as general results, are provided.     

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     

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     

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 augmented orthogonal arrays

Augmented orthogonal arrays (AOAs) were introduced by Stinson, who showed the equivalence between ideal ramp schemes and AOAs (Discrete Math. 341 (2018), 299–307). In this paper, we show that there is an AOA if and only if there is an OA which can be partitioned into subarrays, each being an OA, and that there is a linear AOA if and only if there is a linear maximum distance separable (MDS) code of length and dimension over , which contains a linear MDS subcode of length and dimension over . Some constructions for AOAs and some new infinite classes of AOAs are also given.     

Group Divisible Covering Designs with Block Size 4: A Type of Covering Array with Row Limit

A k‐GDCD, group divisible covering design, of type is a triple , where V is a set of gu elements, is a partition of V into u sets of size g, called groups, and is a collection of k‐subsets of V, called blocks, such that every pair of elements in V is either contained in a unique group or there is at least one block containing it, but not both. This family of combinatorial objects is equivalent to a special case of the graph covering problem and a generalization of covering arrays, which we call CARLs. In this paper, we show that there exists an integer such that for any positive integers g and , there exists a 4‐GDCD of type which in the worst case exceeds the Schönheim lower bound by δ blocks, except maybe when (1) and , or (2) , , and or . To show this, we develop constructions of 4‐GDCDs, which depend on two types of ingredients: essential, which are used multiple times, and auxiliary, which are used only once in the construction. If the essential ingredients meet the lower bound, the products of the construction differ from the lower bound by as many blocks as the optimal size of the auxiliary ingredient differs from the lower bound.     

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.     

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 (aij) (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 Δlh={dh1−dl1,dh2−dl2,…,dhn−dln} contains every element of G at least once.Covering 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 v³+v² when k=5 and v≡2 (mod 4), or k=6, v≡2 (mod 4) and gcd(v,18)≠3.     

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.     

On finding mixed orthogonal arrays of strength 2 with many 2-level factors

We describe a method for finding mixed orthogonal arrays of strength 2 with a large number of 2-level factors. The method starts with an orthogonal array of strength 2, possibly tight, that contains mostly 2-level factors. By a computer search of this starting array, we attempt to find as large a number of 2-level factors as possible that can be used in a new orthogonal array of strength 2 containing one additional factor at more than two levels. The method produces new orthogonal arrays for some parameters, and matches the best-known arrays for others. It is especially useful for finding arrays with one or two factors at more than two levels.     

A generalization of Heffter arrays

In this paper, we define a new class of partially filled arrays, called relative Heffter arrays, that are a generalization of the Heffter arrays introduced by Archdeacon in 2015. Let $v=2nk+t$ be a positive integer, where $t$ divides $2nk$, and let $J$ be the subgroup of ${\mathbb{Z}}_v$ of order $t$. A ${\mathrm{H}}_t\left(m,n;s,k\right)$ Heffter array over ${\mathbb{Z}}_v$ relative to $J$ is an $m\times n$ partially filled array with elements in ${\mathbb{Z}}_v$ such that (a) each row contains $s$ filled cells and each column contains $k$ filled cells; (b) for every $x\in {\mathbb{Z}}_v\backslash J$, either $x$ or $?x$ appears in the array; and (c) the elements in every row and column sum to $0$. Here we study the existence of square integer (i.e., with entries chosen in $\pm \left\{1,\text{…},?\frac{2nk+t}{2}?\right\}$ and where the sums are zero in $\mathbb{Z}$) relative Heffter arrays for $t=k$, denoted by ${\mathrm{H}}_k\left(n;k\right)$. In particular, we prove that for $3\le k\le n$, with $k\ne 5$, there exists an integer ${\mathrm{H}}_k\left(n;k\right)$ if and only if one of the following holds: (a) $k$ is odd and $n\equiv 0,3\left(\text{mod}4\right)$; (b) $k\equiv 2\left(\text{mod}4\right)$ and $n$ is even; (c) $k\equiv 0\left(\text{mod}4\right)$. Also, we show how these arrays give rise to cyclic cycle decompositions of the complete multipartite graph.     

A survey of orthogonal arrays of strength two

ＡＳＵＲＶＥＹＯＦＯＲＴＨＯＧＯＮＡＬＡＲＲＡＹＳＯＦＳＴＲＥＮＧＴＨＴＷＯＬＩＵＺＨＡＮＧＷＥＮ（刘璋温）（ＩｎｓｔｉｔｕｔｅｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ．ｔｈｅＣｈｉｎｅｓｅＡｃａｄｅｍｙｏｆＳｃｉｅｔｉｃｅｓ．Ｂｅｉｊｉｎｇ１０００...