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.     

On t-Covering Arrays

This paper concerns construction methods for t-covering arrays. Firstly, a construction method using perfect hash families is discussed by combining with recursion techniques and error-correcting codes. In particular, by using algebraic-geometric codes for this method we obtain infinite families of t-covering arrays which are proved to be better than currently known probabilistic bounds for covering arrays. Secondly, inspired from a result of Roux [16] and also from a recent result of Chateauneuf and Kreher [6] for 3-covering arrays, we present several explicit constructions for t-covering arrays, which can be viewed as generalizations of their results for t-covering arrays.     

Disjoint skolem sequences and related disjoint structures

A Skolem sequence of order n is a sequence S = (s₁, s₂…, s²ⁿ) of 2n integers satisfying the following conditions: (1) for every k ∈ {1, 2,… n} there exist exactly two elements s_i,S_j such that S_i = S_j = k; (2) If s_i = s_j = k,i < j then j ? i = k. In this article we show the existence of disjoint Skolem, disjoint hooked Skolem, and disjoint near-Skolem sequences. Then we apply these concepts to the existence problems of disjoint cyclic Steiner and Mendelsohn triple systems and the existence of disjoint 1-covering designs. © 1993 John Wiley & Sons, Inc.     

Perfect difference families,perfect difference matrices,and related combinatorial structures

The existence problems of perfect difference families with block size k, k=4,5, and additive sequences of permutations of length n, n=3,4, are two outstanding open problems in combinatorial design theory for more than 30 years. In this article, we mainly investigate perfect difference families with block size k=4 and additive sequences of permutations of length n=3. The necessary condition for the existence of a perfect difference family with block size 4 and order v, or briefly (v, 4,1)‐PDF, is v≡1(mod12), and that of an additive sequence of permutations of length 3 and order m, or briefly ASP (3, m), is m≡1(mod2). So far, (12t+1,4,1)‐PDFs with t<50 are known only for t=1,4−36,41,46 with two definiteexceptions of t=2,3, and ASP (3, m)'s with odd 3<m<200 are known only for m=5,7,13−29,35,45,49,65,75,85,91,95,105,115,119,121,125,133,135,145,147,161,169,175,189,195 with two definite exceptions of m=9,11. In this article, we show that a (12t+1,4,1)‐PDF exists for any t⩽1,000 except for t=2,3, and an ASP (3, m) exists for any odd 3<m<200 except for m=9,11 and possibly for m=59. The main idea of this article is to use perfect difference families and additive sequences of permutations with “holes”. We first introduce the concepts of an incomplete perfect difference matrix with a regular hole and a perfect difference packing with a regular difference leave, respectively. We show that an additive sequence of permutations is in fact equivalent to a perfect difference matrix, then describe an important recursive construction for perfect difference matrices via perfect difference packings with a regular difference leave. Plenty of perfect difference packings with a desirable difference leave are constructed directly. We also provide a general recursive construction for perfect difference packings, and as its applications, we obtain extensive recursive constructions for perfect difference families, some via incomplete perfect difference matrices with a regular hole. Examples of perfect difference packings directly constructed are used as ingredients in these recursive constructions to produce vast numbers of perfect difference families with block size 4. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 415–449, 2010     

The spectrum of additive BIB designs

In this article, the existence of additive BIB designs is discussed with direct and recursive constructions, together with investigation of a property of resolvability. Such designs can be used to construct infinite families of BIB designs. In particular, we obtain a series of B(sⁿ, ts^m, λt (ts^m ? 1) (s^n‐m ? 1)/[2(s^m ? 1)]) for any positive integer λ, such that sⁿ (sⁿ ? 1) λ ≡ 0 (mod s^m (s^m ? 1) and for any positive integer t with 2 ≤ t ≤ s^n‐m, where s is an odd prime power. Connections between additive BIB designs and other combinatorial objects such as multiply nested designs and perpendicular arrays are discussed. A construction of resolvable BIB designs with v = 4k is also proposed. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 235–254, 2007     

Existential closure of block intersection graphs of infinite designs having finite block size and index

In this article we study the n‐existential closure property of the block intersection graphs of infinite t‐(v, k, λ) designs for which the block size k and the index λ are both finite. We show that such block intersection graphs are 2‐e.c. when 2?t?k ? 1. When λ = 1 and 2?t?k, then a necessary and sufficient condition on n for the block intersection graph to be n‐e.c. is that n?min{t, ?(k ? 1)/(t ? 1)? + 1}. If λ?2 then we show that the block intersection graph is not n‐e.c. for any n?min{t + 1, ?k/t? + 1}, and that for 3?n?min{t, ?k/t?} the block intersection graph is potentially but not necessarily n‐e.c. The cases t = 1 and t = k are also discussed. © 2011 Wiley Periodicals, Inc. J Combin Designs 19: 85–94, 2011     

Multiple blocking sets in finite projective spaces and improvements to the Griesmer bound for linear codes

Belov, Logachev and Sandimirov construct linear codes of minimum distance d for roughly 1/q ^k/2 of the values of d < q ^k-1. In this article we shall prove that, for q = p prime and roughly \frac38{\frac{3}{8}}-th’s of the values of d < q ^k-1, there is no linear code meeting the Griesmer bound. This result uses Blokhuis’ theorem on the size of a t-fold blocking set in PG(2, p), p prime, which we generalise to higher dimensions. We also give more general lower bounds on the size of a t-fold blocking set in PG(δ, q), for arbitrary q and δ ≥ 3. It is known that from a linear code of dimension k with minimum distance d < q ^k-1 that meets the Griesmer bound one can construct a t-fold blocking set of PG(k−1, q). Here, we calculate explicit formulas relating t and d. Finally we show, using the generalised version of Blokhuis’ theorem, that nearly all linear codes over \mathbb F_p{{\mathbb F}_p} of dimension k with minimum distance d < q ^k-1, which meet the Griesmer bound, have codewords of weight at least d + p in subcodes, which contain codewords satisfying certain hypotheses on their supports.     

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.     

Cartesian Product of the Universal Covering Property

The paper (Boxer, J. Math. Imaging Vis. 25:159–171, 2006) introduces the important tool of the universal (2,k)-covering property for classifying digital covering spaces. Even though the study of its Cartesian product property is important in expanding the property, it still remains open. In this paper, investigating various properties of a universal (2,k)-covering, we study a Cartesian product of the universal covering property, which plays an important role in classifying Cartesian products of digital coverings.     

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 the transversal Helly numbers of disjoint and overlapping disks

A family of disks is said to have the property T(k) if any k members of the family have a common line transversal. We call a family of unit diameter disks t-disjoint if the distances between the centers are greater than t. We consider for each natural number k≧ 3 the infimum t_k of the distances t for which any finite family of t-disjoint unit diameter disks with the property T(k) has a line transversal. We determine exact values of t₃ and t₄, and give general lower and upper bounds on the sequence t_k, showing that t_k = O(1/k) as k → ∞. In honour of Helge Tverberg’s seventieth birthday Received: 9 June 2005     

Incomplete perfect mendelsohn designs with block size 4 and holes of size 2 and 3

Let v, k, and n be positive integers. An incomplete perfect Mendelsohn design, denoted by k-IPMD(v, n), is a triple (X, Y, ??) where X is a v-set (of points), Y is an n-subset of X, and ?? is a collection of cyclically ordered k-subsets of X (called blocks) such that every ordered pair (a, b) ∈ (X × X)\(Y × Y) appears t-apart in exactly one block of ?? and no ordered pair (a,b) ∈ Y × Y appears in any block of ?? for any t, where 1 ≤ t ≤ k ? 1. In this article, the necessary conditions for the existence of a 4-IPMD(v, n), namely (v ? n) (v ? 3n ? 1) ≡ 0 (mod 4) and v ≥ 3n + 1, are shown to be sufficient for the case n = 3. For the case n = 2, these conditions are sufficient except for v = 7 and with the possible exception of v = 14,15,18,19,22,23,26,27,30. The latter result provides a useful application to the problem relating to the packing of perfect Mendelsohn designs with block size 4. © 1994 John Wiley & Sons, Inc.     

On-line coloring of perfect graphs

Lovász, Saks, and Trotter showed that there exists an on-line algorithm which will color any on-linek-colorable graph onn vertices withO(nlog^(2k–3) n/log^(2k–4) n) colors. Vishwanathan showed that at least (log ^k–1 n/k ^k ) colors are needed. While these remain the best known bounds, they give a distressingly weak approximation of the number of colors required. In this article we study the case of perfect graphs. We prove that there exists an on-line algorithm which will color any on-linek-colorable perfect graph onn vertices withn ^10k/loglogn colors and that Vishwanathan's techniques can be slightly modified to show that his lower bound also holds for perfect graphs. This suggests that Vishwanathan's lower bound is far from tight in the general case.Research partially supported by Office of Naval Research grant N00014-90-J-1206.     

Finding intersection of rectangles by range search

Three related rectangle intersection problems in k-dimensional space are considered: (1) find the intersections of a rectangle with a given set of rectangles, (2) find the intersecting pairs of rectangles as they are inserted into or deleted from an existing set of rectangles, and (3) find the intersecting pairs of a given set of rectangles. By transforming these problems into range search problems, one need not divide the intersection problem into two subproblems, namely, the edge-intersecting problem and the containment problem, as done by many previous studies. Furthermore, this approach can also solve these subproblems separately, if required. For the first problem the running time is O((log n)^2k−1 + s), where s is the number of intersecting pairs of rectangles. For the second problem the time needed to generate and maintain n rectangles is O(n(log n)^2k) and the time for each query is O((log n)^2k−1 + s). For the third problem the total time is O(n log n + n(log n)^2(k−1) + s) for k ≥ 1.     

An approximation of a nonlinear integral equation driven by a function of boundedp-variation

The uniform distance between the solution of a nonlinear equation driven by a functionh with boundedp-variation and its Milstein-type approximation is estimated by δ _n v γ _p (n) v γ _p ² (n), where δ _n =max(t _k −t _k−1 ) is the maximum step size of the approximation on the interval [0,T], γ _p (n)=max υ _p ^1/p (h;[t _k-1,t _k ]), 1 <p < 2, and υ _p (h;[t _k-1,t _k ]) is thep-variation of the functionh on [t _k-1,t _k]. In particular, ifh is a Lipschitz function of order α, then the uniform distance has the bound δ _n ^α for δ_n <1. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius; Vilnius Technical University, Saulétekio 11, 2054 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp. 317–330, July–September, 1999.     

(3,k)-Factor-Critical Graphs and Toughness

 A graph is (r,k)-factor-critical if the removal of any set of k vertices results in a graph with an r-factor (i.e. an r-regular spanning subgraph). Let t(G) denote the toughness of graph G. In this paper, we show that if t(G)≥4, then G is (3,k)-factor-critical for every non-negative integer k such that n+k even, k<2 t(G)−2 and k≤n−7. Revised: September 21, 1998     

Extremal problems concerning Kneser graphs

Let and be two intersecting families of k-subsets of an n-element set. It is proven that | | ≤ (_k−1ⁿ⁻¹) + (_k−1ⁿ⁻¹) holds for , and equality holds only if there exist two points a, b such that {a, b} ∩ F ≠ for all F . For an example showing that in this case max | | = (1−o(1))(_kⁿ) is given. This disproves an old conjecture of Erdös [7]. In the second part we deal with several generalizations of Kneser's conjecture.     

On the distribution of reducible polynomials

Let ℛ _n (t) denote the set of all reducible polynomials p(X) over ℤ with degree n ≥ 2 and height ≤ t. We determine the true order of magnitude of the cardinality |ℛ _n (t)| of the set ℛ _n (t) by showing that, as t → ∞, t ² log t ≪ |ℛ₂(t)| ≪ t ² log t and t ⁿ ≪ |ℛ _n (t)| ≪ t _n for every fixed n ≥ 3. Further, for 1 < n/2 < k < n fixed let ℛ _k,n (t) ⊂ ℛ _n (t) such that p(X) ∈ ℛ _k,n (t) if and only if p(X) has an irreducible factor in ℤ[X] of degree k. Then, as t → ∞, we always have t ^k+1 ≪ |ℛ _k,n (t)| ≪ t ^k+1 and hence |ℛ _n−1,n (t)| ≫ |ℛ _n (t)| so that ℛ _n−1,n (t) is the dominating subclass of ℛ _n (t) since we can show that |ℛ _n (t)∖ℛ _n−1,n (t)| ≪ t ⁿ⁻¹(log t)².On the contrary, if R _n ^s (t) is the total number of all polynomials in ℛ _n (t) which split completely into linear factors over ℤ, then t ²(log t) ⁿ⁻¹ ≪ R _n ^s (t) ≪ t ² (log t) ⁿ⁻¹ (t → ∞) for every fixed n ≥ 2.      

On the L convergence of Lagrange interpolating entire functions of exponential type

Let f: be a continuous, 2π-periodic function and for each n ε let t_n(f; ·) denote the trigonometric polynomial of degree n interpolating f in the points 2kπ/(2n + 1) (k = 0, ±1, …, ±n). It was shown by J. Marcinkiewicz that lim_{n → ∞} ∝₀^2π¦f(θ) − t_n(f θ)¦^p dθ = 0 for every p > 0. We consider Lagrange interpolation of non-periodic functions by entire functions of exponential type τ > 0 in the points kπ/τ (k = 0, ± 1, ± 2, …) and obtain a result analogous to that of Marcinkiewicz.     

Voronoi diagrams of random lines and flats

It is proved that, for any fixedd ≽ 3 and 0 ≤k ≤ d - 1, the expected combinatorial complexity of the Euclidean Voronoi diagram ofn random &-flats drawn independently from the uniform distribution onk-flats intersecting the unit ball in ℝ^d is Ξ(n ^d/(d-k)) asn → ∞. A by-product of the proof is a density transformation for integrating over sets ofd + 1k-flats in ℝ^d