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In computer experiments, it has become a standard practice to select the inputs that spread out as uniformly as possible over the design space. The resulting designs are called space-filling designs and they are undoubtedly desirable…
We consider the optimization problem of constructing a binary orthogonal array (OA) starting from a bigger one, by removing a specified amount of lines. In particular, we develop a genetic algorithm (GA) where the underlying chromosomes are…
We consider the problem of constructing explicit Hitting sets for Combinatorial Shapes, a class of statistical tests first studied by Gopalan, Meka, Reingold, and Zuckerman (STOC 2011). These generalize many well-studied classes of tests,…
Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. Previous efforts for exact algorithms have been unable to avoid structural problems that appear for…
We construct orthogonal arrays OA$_{\lambda} (k,n)$ (of strength two) having a row that is repeated $m$ times, where $m$ is as large as possible. In particular, we consider OAs where the ratio $m / \lambda$ is as large as possible; these…
Random unitaries are useful in quantum information and related fields, but hard to generate with limited resources. An approximate unitary $k$-design is an ensemble of unitaries with an underlying measure over which the average is close to…
In the study of deterministic distributed algorithms it is commonly assumed that each node has a unique $O(\log n)$-bit identifier. We prove that for a general class of graph problems, local algorithms (constant-time distributed algorithms)…
In an early paper, He and Tang [Biometrika 100 (2013) 254-260] introduced and studied a new class of designs, strong orthogonal arrays, for computer experiments, and characterized such arrays through generalized orthogonal arrays. The…
The applications of random quantum circuits range from quantum computing and quantum many-body systems to the physics of black holes. Many of these applications are related to the generation of quantum pseudorandomness: Random quantum…
The generalized word length pattern of an orthogonal array allows a ranking of orthogonal arrays in terms of the generalized minimum aberration criterion (Xu and Wu [Ann. Statist. 29 (2001) 1066-1077]). We provide a statistical…
Mixed (asymmetric) orthogonal arrays (MOAs) generalize classical orthogonal arrays by allowing columns over different alphabets. However, their study requires very different structural tools than those used for symmetric orthogonal arrays…
In 1987 Hiroshi Maehara conjectured that a graph can be represented by vectors considered adjacent when not orthogonal (a faithful orthogonal representation) in codimension the minimum degree of the graph. Without settling the conjecture,…
A $t$-dimensional orthogonal representation of a hypergraph is an assignment of nonzero vectors in $\mathbb{R}^t$ to its vertices, such that every hyperedge contains two vertices whose vectors are orthogonal. The orthogonality dimension of…
Complex orthogonal designs (CODs) are used to construct space-time block codes. COD $\mathcal{O}_z$ with parameter $[p, n, k]$ is a $p \times n$ matrix, where nonzero entries are filled by $\pm z_i$ or $\pm z^*_i$, $i = 1, 2,..., k$, such…
Explorable heap selection is the problem of selecting the $n$th smallest value in a binary heap. The key values can only be accessed by traversing through the underlying infinite binary tree, and the complexity of the algorithm is measured…
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…
We prove new upper and lower bounds for the Online Orthogonal Vectors Problem ($\mathsf{OnlineOV}_{n,d}$). In this problem, a preprocessing algorithm receives $n$ vectors $x_1,\ldots,x_n\in\{0,1\}^d$ and constructs a data structure of size…
For constrained, not necessarily monotone submodular maximization, all known approximation algorithms with ratio greater than $1/e$ require continuous ideas, such as queries to the multilinear extension of a submodular function and its…
We derive new estimates of the size of codes and orthogonal arrays in the ordered Hamming space (the Niederreiter-Rosenbloom-Tsfasman space). We also show that the eigenvalues of the ordered Hamming scheme, the association scheme that…
We establish new correlation bounds and pseudorandom generators for a collection of computation models. These models are all natural generalizations of structured low-degree $F_2$-polynomials that we did not have correlation bounds for…