Covering and radius-covering arrays: Constructions and classification

Colbourn, Charles J.; Kéri, Gerzson; Soriano, P; Schlage-Puchta, Jan Christoph

doi:10.1016/j.dam.2010.03.008

Cited by 51 publications

(72 citation statements)

References 34 publications

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“…For any i ∈ r , form a binary matrix C i ∈ [2] (m r)×n by choosing each entry independently according to a Bernoulli distribution such that the probability of choosing . Let π i denote the probability that a fixed row of C i denoted by r satisfies conditions (10)- (12). Note that since the entries of C i are chosen according to an i.i.d.…”

Section: Construction Of Binary Sq-separable Codesmentioning

confidence: 99%

“…In its full generality, GT may be viewed as the problem of inferring the state of a system from the superposition of the state vectors of a subset of the system's elements. As such, GT has found many applications in communication theory [2]- [5], signal processing [6]- [8], computer science [9]- [11], and mathematics [12]. Some examples of these applications include error-correcting coding [4], [13], [14], identifying users accessing a multiple access channel (MAC) [15], [16], reconstructing sparse signals from low-dimensional projections [6], [7], and many others.…”

Section: Introductionmentioning

confidence: 99%

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Semiquantitative Group Testing

Emad

Milenković

2014

IEEE Trans. Inform. Theory

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We propose a novel group testing method, termed semi-quantitative group testing, motivated by a class of problems arising in genome screening experiments. Semi-quantitative group testing (SQGT) is a (possibly) non-binary pooling scheme that may be viewed as a concatenation of an adder channel and an integer-valued quantizer. In its full generality, SQGT may be viewed as a unifying framework for group testing, in the sense that most group testing models are special instances of SQGT. For the new testing scheme, we define the notion of SQ-disjunct and SQ-separable codes, representing generalizations of classical disjunct and separable codes. We describe several combinatorial and probabilistic constructions for such codes. While for most of these constructions we assume that the number of defectives is much smaller than total number of test subjects, we also consider the case in which there is no restriction on the number of defectives and they may be as large as the total number of subjects. For the codes constructed in this paper, we describe a number of efficient decoding algorithms. In addition, we describe a belief propagation decoder for sparse SQGT codes for which no other efficient decoder is currently known. Finally, we define the notion of capacity of SQGT and evaluate it for some special choices of parameters using information theoretic methods. The authors are with the

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Section: Construction Of Binary Sq-separable Codesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Semiquantitative Group Testing

Emad

Milenković

2014

IEEE Trans. Inform. Theory

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“…1. The application of the construction T to this base covering array using this matrix δ produces the covering array CA (8,3,4,2).…”

Section: Introductionmentioning

confidence: 99%

“…In this work we propose a method to generate covering arrays based on the construction of a Tower of Covering Arrays (TCA). (4; 2, 3, 2) and to the matrix δ shown in the figure, produces the covering array of strength three CA(8;3,4,2).…”

Section: Introductionmentioning

confidence: 99%

Tower of covering arrays

Torres-Jiménez

Izquierdo-Marquez

Kacker

et al. 2015

Discrete Applied Mathematics

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a b s t r a c tCovering arrays are combinatorial objects that have several practical applications, specially in the design of experiments for software and hardware testing. A covering array of strength t and order v is an N ×k array over Z v with the property that every N ×t subarray covers all members of Z t v at least once. In this work we explore the construction of a Tower of Covering Arrays (TCA) as a way to produce covering arrays that improve or match some current upper bounds. A TCA of height h is a succession of h + 1 covering arrays C 0 , C 1 , . . . , C h in which for i = 1, 2, . . . , h the covering array C i is one unit greater in the number of factors and the strength of the covering array C i−1 ; this way, if the covering array C 0 is of strength t and has k factors then the covering arrays C 1 , . . . , C h are of strength t + 1, . . . , t + h and have k + 1, . . . , k + h factors respectively. We note that the ratio between the number of rows of the last covering array C h in a TCA of height h and the number of rows of the best known covering array for the same values of t, k, and v as for C h is reduced as h grows. Therefore, we search for TCAs with the greatest height possible. The relevant results are the improvement of nineteen current upper bounds for v = 2 and t ∈ {7, 8, 9, 10, 11}, and the construction of twenty-one covering arrays that matched current upper bounds.

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“…In [27], the authors detail a technique called fusion that permits reducing the number of levels by one in a covering array while allowing two rows to be dropped; this is a generalization from [26]. Theorem 2.2.15 (Fusion, [27]).…”

Section: Bounds On Canmentioning

confidence: 99%

Variable strength covering arrays

Raaphorst

Moura

Stevens

2018

J of Combinatorial Designs

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Recently, covering arrays have been the subject of considerable research attention as they hold both theoretical interest and practical importance due to their applications to testing. In this thesis, we perform the first comprehensive study of a generalization of covering arrays called variable strength covering arrays, where we dictate the interactions to be covered in the array by modeling them as facets of an abstract simplicial complex.We outline the necessary background in the theory of hypergraphs, combinatorial testing, and design theory that is relevant to the study of variable strength covering arrays. We then approach questions that arise in variable strength covering arrays in a number of ways. We demonstrate their connections to hypergraph homomorphisms, and explore the properties of a particular family of abstract simplicial complexes, the qualitative independence hypergraphs. These hypergraphs are tightly linked to variable strength covering arrays, and we determine and identify several of their important properties and subhypergraphs.We give a detailed study of constructions for variable strength covering arrays, and provide several operations and divide-and-conquer techniques that can be used in building them. In addition, we give a construction using linear feedback shift registers from primitive polynomials of degree 3 over arbitrary finite fields to find variable strength covering arrays, which we extend to strength-3 covering arrays whose sizes are smaller than many of the best known sizes of covering arrays.We then give an algorithm for creating variable strength covering arrays over arbitrary abstract simplicial complexes, which builds the arrays one row at a time, using a density concept to guarantee that the size of the resultant array is asymptotic in the logarithm of the number of facets in the abstact simplicial complex. This algorithm is of immediate practical importance, as it can be used to create test suites for combinatorial testing.Finally, we use the Lovász Local Lemma to nonconstructively determine upper bounds on the sizes of arrays for a number of different families of hypergraphs. We lay out a framework that can be used for many hypergraphs, and then discuss possible strategies that can be taken in asymmetric problems.ii Acknowledgements This thesis certainly could not have been completed without the help of many people.Firstly and most importantly, I would like to extend my immeasurable thanks to both of my supervisors, Lucia Moura and Brett Stevens. As a lowly undergraduate student many years ago, I had the extreme fortune of being chosen to serve as a teaching assistant to work with Lucia Moura. I could never have predicted that that was to be the first step on a long and rewarding journey into academia. Through all the opportunities she gave me, I caught her infectious enthusiasm for research and had my eyes opened to the wonderful fields of design theory and covering arrays. Without her endless patience and compassion with my quirkiness, this thesis would never...

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Covering and radius-covering arrays: Constructions and classification

Cited by 51 publications

References 34 publications

Semiquantitative Group Testing

Semiquantitative Group Testing

Tower of covering arrays

Variable strength covering arrays

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