An $(n,k)$ sequence covering array is a set of permutations of $[n]$ such that each sequence of $k$ distinct elements of $[n]$ is a subsequence of at least one of the permutations. An $(n,k)$ sequence covering array is perfect if there is a positive integer $λ$ such that each sequence of $k$ distinct elements of $[n]$ is a subsequence of precisely $λ$ of the permutations.
While relatively close upper and lower bounds for the minimum size of a sequence covering array are known, this is not the case for perfect sequence covering arrays. Here we present new nontrivial bounds for the latter. In particular, for $k=3$ we obtain a linear lower bound and an almost linear upper bound.
It has been conjectured that whenever an optimal covering array exists there is also a uniform covering array with the same parameters and this is true for all known optimal covering arrays. When used as a test suite, the application context may have pairs of parameters that must be avoided and Covering arrays avoiding forbidden edges (CAFE) are a generalization accommodating this requirement. We prove that there is an arc-transitive, highly symmetric constraint graph where the unique optimal covering array avoiding forbidden edges is not uniform. This does not refute the conjecture but it doe…
A covering array $\rm{CA}(N;t,k,v)$ 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 \emph{optimal} if it has the smallest possible $N$ for given $t$, $k$, and $v$, and \emph{uniform} if every symbol occurs $\lfloor N/v \rfloor$ or $\lceil N/v \rceil$ times in every column. Prior to 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 Erdős-Ko-Rado Theorem, a…
Let G be a bridgeless cubic graph. A well-known conjecture of Berge and Fulkerson can be stated as follows: there exist five perfect matchings of G such that each edge of G is contained in at least one of them. Here, we prove that in each bridgeless cubic graph there exist five perfect matchings covering a portion of the edges at least equal to 215/231 . By a generalization of this result, we decrease the best known upper bound, expressed in terms of the size of the graph, for the number of perfect matchings needed to cover the edge-set of G.
The main focus of this thesis is a generalization of covering arrays, covering arrays on graphs. Two vectors v,w in Z_k^n are qualitatively independent if for all ordered pairs (a,b) in Z_k x Z_k there is a position i in the vectors where (a,b) = (v_i,w_i). A covering array is an array with the property that any pair of rows are qualitatively independent. A covering array on a graph is an array with a row for each vertex of the graph with the property that any two rows which correspond to adjacent vertices are qualitatively independent. The addition of a graph structure to covering arrays make…
Berge Conjecture states that every bridgeless cubic graph has 5 perfect matchings such that each edge is contained in at least one of them. In this paper, we show that Berge Conjecture holds for two classes of cubic graphs, cubic graphs with a circuit missing only one vertex and bridgeless cubic graphs with a 2-factor consisting of two circuits. The first part of this result implies that Berge Conjecture holds for hypohamiltonian cubic graphs.
A $q$-ary $t$-covering array is an $m \times n$ matrix with entries from $\{0, 1, ..., q-1\}$ with the property that for any $t$ column positions, all $q^t$ possible vectors of length $t$ occur at least once. One wishes to minimize $m$ for given $t$ and $n$, or maximize $n$ for given $t$ and $m$. For $t = 2$ and $q = 2$, it is completely solved by Rényi, Katona, and Kleitman and Spencer. They also show that maximal binary 2-covering arrays are uniquely determined. Roux found the lower bound of $m$ for a general $t, n$, and $q$. In this article, we show that $m \times n$ binary 2-covering array…
Consider the family of all perfect matchings of the complete graph $K_{2n}$ with $2n$ vertices. Given any collection $\mathcal M$ of perfect matchings of size $s$, there exists a maximum number $f(n,x)$ such that if $s\leq f(n,x)$, then there exists a perfect matching that agrees with each perfect matching in $\mathcal M$ in at most $x-1$ edges. We use probabilistic arguments to give several lower bounds for $f(n,x)$. We also apply the Lovász local lemma to find a function $g(n,x)$ such that if each edge appears at most $g(n, x)$ times then there exists a perfect matching that agrees with each…
The focus of this thesis is the study and construction of covering arrays, relying on maximal period sequences and other tools from finite fields. A covering array of strength $t$, denoted $\mathrm{CA}(N; t, k,v)$, is an $N\times k$ array with entries from an alphabet $A$ of size $v$, with the property that in the $N\times t$ subarray defined by any $t$ columns, each of the $v^t$ vectors in $A^t$ appears at least once as a row. Covering arrays generalize orthogonal arrays, which are classic combinatorial objects that have been studied extensively. Constructing covering arrays with a small row-…
Covering arrays are generalizations of orthogonal arrays that have been widely studied and are used in software testing. The probabilistic method has been employed to derive upper bounds on the sizes of minimum covering arrays and give asymptotic upper bounds that are logarithmic on the number of columns of the array. This corresponds to test suites with a desired level of coverage of the parameter space where we guarantee the number of test cases is logarithmic on the number of parameters of the system. In this paper, we study variable strength covering arrays, a generalization of covering ar…
Covering arrays for words of length $t$ over a $d$ letter alphabet are $k \times n$ arrays with entries from the alphabet so that for each choice of $t$ columns, each of the $d^t$ $t$-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 words are equivalent if they induce the same partition of a $t$ 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 $k=k(n)$ of a covering array. Definitiv…
A $t$-covering array with entries from the alphabet ${\cal Q}=\{0,1,\ldots,q-1\}$ is a $k\times n$ stack, so that for any choice of $t$ (typically non-consecutive) columns, each of the $q^{t}$ possible $t$-letter words over ${\cal Q}$ appear at least once among the rows of the selected columns. We will show how a combination of the Lovász local lemma; combinatorial analysis; Stirling's formula; and Calculus enables one to find better asymptotic bounds for the minimum size of $t$-covering arrays, notably for $t = 3, 4$. Here size is measured in the number of rows, as expressed in terms of the n…
Difference arrays are used in applications such as software testing, authentication codes and data compression. Pseudo-orthogonal Latin squares are used in experimental designs. A special class of pseudo-orthogonal Latin squares are the mutually nearly orthogonal Latin squares (MNOLS) first discussed in 2002, with general constructions given in 2007. In this paper we develop row complete MNOLS from difference covering arrays. We will use this connection to settle the spectrum question for sets of 3 mutually pseudo-orthogonal Latin squares of even order, for all but the order 146.
We study perfect multiple coverings in translation invariant graphs with vertex set $\mathbb{Z}^2$ using an algebraic approach. In this approach we consider any such covering as a two-dimensional binary configuration which we then express as a two-variate formal power series. Using known results, we conclude that any perfect multiple covering has a non-trivial periodizer, that is, there exists a non-zero polynomial whose formal product with the power series presenting the covering is a two-periodic configuration. If a non-trivial periodizer has line polynomial factors in at most one direction,…
A covering array $CA(N; t,k,v)$ is an $N \times k$ array $A$ whose each cell takes a value for a $v$-set $V$ called an alphabet. Moreover, the set $V^t$ is contained in the set of rows of every $N \times t$ subarray of $A$. The parameter $N$ is called the size of an array and $CAN(t,k,v)$ denotes the smallest $N$ for which a $CA(N; t,k,v)$ exists. It is well known that $CAN(t,k,v) = {\rm Θ}(\log_2 k)$~\cite{godbole_bounds_1996}. In this paper we derive two upper bounds on $d(t,v)=\limsup_{k \rightarrow \infty} \frac{CAN(t,k,v)}{\log_2 k}$ using the algorithmic approach to the Lovász local lemm…
Covering arrays find important application in software and hardware interaction testing. For practical applications it is useful to determine or bound the minimum number of rows, CAN$(t,k,v)$, in a covering array for given values of the parameters $t,k$ and $v$. Asymptotic upper bounds for CAN$(t,k,v)$ have earlier been established using the Stein-Lovász-Johnson strategy and the Lovász local lemma. A series of improvements on these bounds is developed in this paper. First an estimate for the discrete Stein-Lovász-Johnson bound is derived. Then using alteration, the Stein-Lovász-Johnson bound i…
A covering array CA(N; t; k; v) is an N x k array on v symbols such that every N x t subarray contains as a row each t-tuple over the v symbols at least once. The minimum N for which a CA(N; t; k; v) exists is called the covering array number of t, k, and v, and it is denoted by CAN(t; k; v). In this work we prove new CANs using an optimized procedure.
In a recent survey, Schmidt compiled equivalences between generalized bent functions, group invariant Butson Hadamard matrices, and abelian splitting relative difference sets. We establish a broader network of equivalences by considering Butson matrices that are cocyclic rather than strictly group invariant. This result has several applications; for example, to the construction of Boolean functions whose expansions are generalized partially bent functions, including cases where no bent function can exist.
This work shows several direct and recursive constructions of ordered covering arrays using projection, fusion, column augmentation, derivation, concatenation and cartesian product. Upper bounds on covering codes in NRT spaces are also obtained by improving a general upper bound. We explore the connection between ordered covering arrays and covering codes in NRT spaces, which generalize similar results for the Hamming metric. Combining the new upper bounds for covering codes in NRT spaces and ordered covering arrays, we improve upper bounds on covering codes in NRT spaces for larger alphabets.…
An $(n,k)$-perfect sequence covering array with multiplicity $λ$, denoted PSCA$(n,k,λ)$, is a multiset whose elements are permutations of the sequence $(1,2, \dots, n)$ and which collectively contain each ordered length $k$ subsequence exactly $λ$ times. The primary objective is to determine for each pair $(n,k)$ the smallest value of $λ$, denoted $g(n,k)$, for which a PSCA$(n,k,λ)$ exists; and more generally, the complete set of values $λ$ for which a PSCA$(n,k,λ)$ exists. Yuster recently determined the first known value of $g(n,k)$ greater than 1, namely $g(5,3)=2$, and suggested that findin…
A covering array $t$-$CA(n,k,g)$, of size $n$, strength $t$, degree $k$, and order $g$, is a $k\times n$ array on $g$ symbols such that every $t\times n$ sub-array contains every $t\times 1$ column on $g$ symbols at least once. Covering arrays have been studied for their applications to software testing, hardware testing, drug screening, and in areas where interactions of multiple parameters are to be tested. In this paper, we present an algebraic construction that improves many of the best known upper bounds on $n$ for covering arrays 4-$CA(n,k,g)$ with $g=3$. The $coverage$ $measure$ $μ_t(A)…
Two vectors $x,y$ in $\mathbb{Z}_g^n$ are $ qualitatively$ $ independent$ if for all pairs $(a,b)\in \mathbb{Z}_g\times \mathbb{Z}_g$, there exists $i\in \{1,2,\ldots,n\}$ such that $(x_i,y_i)=(a,b)$. A covering array on a graph $G$, denoted by $CA(n,G,g)$, is a $|V(G)|\times n$ array on $\mathbb{Z}_g$ with the property that any two rows which correspond to adjacent vertices in $G$ are qualitatively independent. The number of columns in such array is called its $size$. Given a graph $G$, a covering array on $G$ with minimum size is called $optimal$. Our primary concern in this paper is with co…
The classical Erd\H os-Ko-Rado theorem states that if $k\le\floor{n/2}$ then the largest family of pairwise intersecting $k$-subsets of $[n]=\{0,1,...,n\}$ is of size ${{n-1}\choose{k-1}}$. A family of $k$ subsets satisfying this pairwise intersecting property is called an EKR family. We generalize the EKR property and provide asymptotic lower bounds on the size of the largest family ${\cal A}$ of $k$-subsets of $[n]$ that satisfies the following property: For each $A,B,C\in{\cal A}$, each of the four sets $A\cap B\cap C;A\cap B\cap C^C; A\cap B^C\cap C; A^C\cap B\cap C$ are non-empty. This ge…
In this work, constructions of ordered covering arrays are discussed and applied to obtain new upper bounds on covering codes in Rosenbloom-Tsfasman spaces (RT spaces), improving or extending some previous results.
Covering arrays are combinatorial objects that have been successfully applied in the design of test suites for testing systems such as software, circuits and networks, where failures can be caused by the interaction between their parameters. In this paper, we perform a new generalization of covering arrays called covering arrays on 3-uniform hypergraphs. Let $n, k$ be positive integers with $k\geq 3$. Three vectors $x\in \mathbb Z_{g_1}^n$, $y\in \mathbb Z_{g_2}^n$, $z\in \mathbb Z_{g_3}^n$ are {\it 3-qualitatively independent} if for any triplet $(a, b, c) \in \mathbb Z_{g_1}\,\times\, \mathb…
Let $G$ be a finite abelian group and let $\exp(G)$ denote the least common multiple of the orders of all elements of $G$. A $BH(G,h)$ matrix is a $G$-invariant $|G|\times |G|$ matrix $H$ whose entries are complex $h$th roots of unity such that $HH^*=|G|I_{|G|}$. In this paper, we study the relation between $G$ and $h$ so that a $BH(G,h)$ matrix exists. We will only focus on $BH(\mathbb{Z}_n,h)$ matrices and $BH(G,2p^b)$ matrices, where $p$ is an odd prime. By our results, there are $2687$ open cases left for the existence of $BH(\mathbb{Z}_n,h)$ matrices in which $1\leq n,h \leq 100$. In the …
Given a set $S$ of $v \ge 2$ symbols, and integers $k \ge t \ge 2$ and $N \ge 1$, an $N \times k$ array $A \in S^{N \times k}$ is an $(N; t, k, v)$-covering array if all sequences in $S^t$ appear as rows in every $N \times t$ subarray of $A$. These arrays have a wide variety of applications, driving the search for small covering arrays. The covering array number, $\mathrm{CAN}(t,k,v)$, is the smallest $N$ for which an $(N; t,k,v)$-covering array exists. In this paper, we combine probabilistic and linear algebraic constructions to improve the upper bounds on $\mathrm{CAN}(t,k,v)$ by a factor of…
A covering array $\mathsf{CA}(N;t,k,v)$ is an $N\times k$ array with entries in $\{1, 2, \ldots , v\}$, for which every $N\times t$ subarray contains each $t$-tuple of $\{1, 2, \ldots , v\}^t$ 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 $\mathsf{CAN}(t,k,v)$, the minimum number $N$ of rows of a $\mathsf{CA}(N;t,k,v)$. The well known bound $\mathsf{CAN}(t,k,v)=O((t-1)v^t\log k)$ is not too far from being asymptotically optimal.…
A $t-\a$ covering array is an $m\times n$ matrix, with entries from an alphabet of size $α$, such that for any choice of $t$ rows, and any ordered string of $t$ letters of the alphabet, there exists a column such that the "values" of the rows in that column match those of the string of letters. We use the Lovász Local Lemma in conjunction with a new tiling-based probability model to improve the upper bound on the smallest number of columns $N=N(m,t,α)$ of a $t-\a$ covering array.
Generalized perfect binary arrays (GPBAs) were used by Jedwab to construct perfect binary arrays. A non-trivial GPBA can exist only if its energy is $2$ or a multiple of $4$. This paper introduces generalized optimal binary arrays (GOBAs) with even energy not divisible by $4$, as analogs of GPBAs. We give a procedure to construct GOBAs based on a characterization of the arrays in terms of $2$-cocycles. As a further application, we determine negaperiodic Golay pairs arising from generalized optimal binary sequences of small length.
We introduce two-dimensional toroidal arrays that are a variant of the de Bruijn tori. We call them nested perfect toroidal arrays. Instead of asking that every array of a given size has exactly one occurrence, we partition the positions in congruence classes and we ask exactly one occurrence in each congruence class. We also ask that this property applies recursively to each of the subarrays. We give a method to construct nested perfect toroidal arrays based on Pascal triangle matrix modulo 2. For the two-symbol alphabet, and for $n$ being a power of $2$, our method yields $2^{n^2+n-1}$ diffe…
We consider the problem of finding a set (partial covering array) $S$ of vertices of the Boolean $n$-cube having cardinality $2^{n-k}$ and intersecting with maximum number of $k$-dimensional faces. We prove that the ratio between the numbers of the $k$-faces containing elements of $S$ to $k$-faces is less than $1-\frac{1+o(1)}{\sqrt{2πk}}$ as $n\rightarrow\infty$ for sufficiently large $k$. The solution of the problem in the class of linear codes is found. Connections between this problem, cryptography and an efficiency of quantization are discussed.
A PSCA$(v, t, λ)$ is a multiset of permutations of the $v$-element alphabet $\{0, \dots, v-1\}$ such that every sequence of $t$ distinct elements of the alphabet appears in the specified order in exactly $λ$ permutations. For $v \geq t$, let $g(v, t)$ be the smallest positive integer $λ$ such that a PSCA$(v, t, λ)$ exists. We present an explicit construction that proves $g(v,t) = O(v^{t(t-2)})$ for fixed $t \geq 4$. The method of construction involves taking a permutation representation of the group of projectivities of a suitable projective space of dimension $t - 2$ and deleting all but a ce…
This paper presents a combinatorial construction of low-density parity-check (LDPC) codes from difference covering arrays. While the original construction by Gallagher was by randomly allocating bits in a sparse parity-check matrix, over the past 20 years researchers have used a variety of more structured approaches to construct these codes, with the more recent constructions of well-structured LDPC coming from balanced incomplete block designs (BIBDs) and from Latin squares over finite fields. However these constructions have suffered from the limited orders for which these designs exist. Her…
Given $3 \leq k \leq s$, we say that a $k$-uniform hypergraph $C^k_s$ is a tight cycle on $s$ vertices if there is a cyclic ordering of the vertices of $C^k_s$ such that every $k$ consecutive vertices under this ordering form an edge. We prove that if $k \ge 3$ and $s \ge 2k^2$, then every $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2 + o(1))n$ has the property that every vertex is covered by a copy of $C^k_s$. Our result is asymptotically best possible for infinitely many pairs of $s$ and $k$, e.g. when $s$ and $k$ are coprime. A perfect $C^k_s$-tiling is a spann…
Modern software systems often consist of many different components, each with a number of options. Although unit tests may reveal faulty options for individual components, functionally correct components may interact in unforeseen ways to cause a fault. Covering arrays are used to test for interactions among components systematically. A two-stage framework, providing a number of concrete algorithms, is developed for the efficient construction of covering arrays. %Our framework divides the construction in two stages. In the first stage, a time and memory efficient randomized algorithm covers mo…
We give a framework for generalizing LDPC code constructions that use Transversal Designs or related structures such as mutually orthogonal Latin squares. Our construction offers a broader range of code lengths and codes rates. Similar earlier constructions rely on the existence of finite fields of order a power of a prime. In contrast the LDPC codes constructed here are based on difference matrices and difference covering arrays, structures available for any order $a$. They satisfy the RC constraint and have, for $a$ odd, length $a^2$ and rate $1-\frac{4a-3}{a^2}$, and for $a$ even, length $a…
With a computer-aided approach based on the connection with equitable partitions, we establish the uniqueness of the orthogonal array OA$(1536,13,2,7)$, constructed in [D.G.Fon-Der-Flaass. Perfect $2$-Colorings of a Hypercube, Sib. Math. J. 48 (2007), 740-745] as an equitable partition of the $13$-cube with quotient matrix $[[0,13],[3,10]]$. By shortening the OA$(1536,13,2,7)$, we obtain $3$ inequivalent orthogonal arrays OA$(768,12,2,6)$, which is a complete classification for these parameters too. After our computing, the first parameters of unclassified binary orthogonal arrays OA$(N,n,2,t)…
We present new constructions for perfect and odd perfect sequences over the quaternion group $Q_8$. In particular, we show for the first time that perfect and odd perfect quaternion sequences exist in all lengths $2^t$ for $t\geq0$. In doing so we disprove the quaternionic form of Mow's conjecture that the longest perfect $Q_8$-sequence that can be constructed from an orthogonal array construction is of length 64. Furthermore, we use a connection to combinatorial design theory to prove the existence of a new infinite class of Williamson sequences, showing that Williamson sequences of length $2…
