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A covering array CA(N;t, k, v) is an N × k array with entries in {1,2,…, v}, for which everyN × t subarray contains each t-tuple of {1,2,…, v}t among its rows. Covering arrays find application in interaction testing, including software and hardware testing, advanced materials development, and biological systems. A central question is to determine or bound CAN(t, k, v), the minimum number N of rows of a CA(N;t, k, v). The well known bound CAN(t, k, v) = O((t − 1)vt log k) is not too far from being asymptotically optimal. Sensible relaxations of the covering requirement arise when (1) the set {1,2,…, v}t need only be contained among the rows of at least(1−휖)kt of the N × t subarrays and (2) the rows of everyN × t subarray need only contain a (large) subset of {1,2,…, v}t. In this paper, using probabilistic methods, significant improvements on the covering array upper bound are established for both relaxations, and for the conjunction of the two. In each case, a randomized algorithm constructs such arrays in expected polynomial time. [ABSTRACT FROM AUTHOR]

We use deterministic and probabilistic methods to analyze the performance of compressed sensing matrices constructed from Hadamard matrices and pairwise balanced designs, previously introduced by a subset of the authors. In this paper, we obtain upper and lower bounds on the sparsity of signals for which our matrices guarantee recovery. These bounds are tight to within a multiplicative factor of at most 4\sqrt 2 . We provide new theoretical results and detailed simulations, which indicate that the construction is competitive with Gaussian random matrices, and that recovery is tolerant to noise. A new recovery algorithm tailored to the construction is also given. [ABSTRACT FROM AUTHOR]

Necessary conditions for existence of a resolvable group divisible design (GDD) with block size 3 and type [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A known recursive construction for perfect hash families multiplies the number of factors by q, where q is the largest prime power not exceeding the number of symbols. This is generalized to permit multiplication of the number of factors by powers of q; it is also improved upon by restricting the structure of the ingredient designs. For a perfect hash family of strength t, it may be possible to order its rows so that, for each , the first Ni rows form a perfect hash family of strength . This 'matroshka' ordering underlies a sometimes substantial reduction in the number of rows generated by the recursive construction. Perfect hash families with useful matroshka orderings are generated using linear orthogonal arrays, and with an effective one-row-at-a-time greedy algorithm. [ABSTRACT FROM AUTHOR]

Abstract: The existence of graph designs for the two nonisomorphic graphs on five vertices and eight edges is determined in the case of index one, with three possible exceptions in total. It is established that for the unique graph with vertex sequence (3, 3, 3, 3, 4), a graph design of order exists exactly when and , with the possible exception of . For the unique graph with vertex sequence , a graph design of order exists exactly when , with the possible exceptions of . [Copyright &y& Elsevier]

Abstract: A covering array is an array such that every sub-array contains all t-tuples from 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. [Copyright &y& Elsevier]

Abstract: A covering array is an array such that every sub-array contains all t-tuples from symbols at least once, where t is the strength of the array. Covering arrays are used in experiments to screen for interactions among t-subsets of k components. Strength two covering arrays have been studied from numerous viewpoints, resulting in a variety of computational, direct, and recursive constructions. Consequently, it can be difficult to determine the smallest covering array that results from known construction. To address this, existence tables for the best currently known covering arrays are presented. In the process, a new direct construction from orthogonal arrays is also introduced. [Copyright &y& Elsevier]

SONET/WDM networks using wavelength add-drop multiplexing can be constructed using certain graph decompositions used to form a grooming, consisting of unions of primitive rings. The cost of such a decomposition is the sum, over all graphs in the decomposition, of the number of vertices of nonzero degree in the graph. The existence of such decompositions with minimum cost, when every pair of sites employs no more than $\frac{1}{6}$ of the wavelength capacity, is determined with a finite number of possible exceptions. Indeed, when the number N of sites satisfies $N \equiv 1 \pmod{3}$, the determination is complete, and when $N \equiv 2 \pmod{3}$, the only value left undetermined is N = 17. When $N \equiv 0 \pmod{3}$, a finite number of values of N remain, the largest being N = 2580. The techniques developed rely heavily on tools from combinatorial design theory. [ABSTRACT FROM AUTHOR]

A difference triangle set (D$\Delta$S) is a collection of sets of integers having the property that every integer can be written in at most one way as the difference of two elements within a set of the collection. The standard objective is to minimize the largest difference represented, given a specified size of the collection and sizes of the sets that it contains. In order to construct D$\Delta$Ss, we present a new type of combinatorial design, monotonic directed $(v,k,\lambda)$- designs (MDDs). Using MDDs, we give a general recursive construction for difference triangle sets (D$\Delta$Ss). Several instances of this main construction are derived. One of these, the perfect construction, leads to an infinite family of regular (optimal) D$\Delta$Ss if the existence of a single regular D$\Delta$S is known. [ABSTRACT FROM AUTHOR]

In a synchronous optical network ring, assignment of source-to-destination circuits to wavelengths must respect traffic requirements and minimize both the number of wavelengths and the amount of terminal conversion equipment. When traffic requirements are approximately equal on all source–destination circuits, the assignment can be modeled as a graph decomposition problem. In this setting, techniques from combinatorial design theory can be applied. These techniques are introduced in a simpler form when every source–destination circuit requires one quarter of a wavelength. More sophisticated design-theoretic methods are then developed to produce the required decompositions for all sufficiently large ring sizes, when each source–destination circuit requires one eighth of a wavelength. [Copyright &y& Elsevier]

Explores the role of combinatorial designs in solving problems that are basic to the field of computing. Case studies of the applications of designs to computation; Areas where graph-theoretic tools are used.