arXiv subjects

Guangzhou Chen

Publications and source records attributed to Guangzhou Chen.

8 recordsLinked to original sources

arXiv2026

Improved Bounds for Nested Orthogonal Arrays

Nested orthogonal arrays (NOAs) have found increasing application in various experimental design problems. A central challenge in this field is the derivation of lower bounds on the number of runs. These bounds serve as a powerful criterion to prove the nonexistence of specific arrays. For symmetric NOAs, Mukerjee, Qian, and Wu developed statistical arguments that yield fairly tight bounds. By contrast, the bounds for asymmetric NOAs proposed by Lin, Pang and Chen, which are obtained via a recursive column deletion technique that reduces the general problem to NOAs of strength 2, are not optimal. Consequently, improving these bounds remains a significant open problem. In this paper, we reformulate the Rao bound and establish a new bound for NOAs under a group theoretic framework. Using the character theory of finite abelian groups, we obtain an equivalent characterization via group characters. This framework allows us to give a new proof of Rao bound for orthogonal arrays and to derive significantly sharper lower bounds for asymmetric NOAs than those of Lin et al. In the case where all factor levels are equal, our bounds reduce naturally to the symmetric bounds of Mukerjee, Qian, and Wu. We also confirm their optimality by explicitly two constructions of NOAs that achieve these bounds.

math.CO↗

arXiv2026

Grouped Orthogonal Arrays from Orthogonal Arrays and Difference Schemes

Grouped orthogonal arrays were introduced to address experimental design problems arising in computer experiments with grouped inputs, as well as in physical experiments where interactions between factors from different groups are assumed to be negligible. Motivated by the growing need for flexible and efficient designs under such settings, this article develops several constructions to expand the existing catalogs of grouped orthogonal arrays. The proposed constructions provide a large collection of new grouped orthogonal arrays with significantly larger numbers of groups and group sizes.

stat.ME↗

arXiv2025

Grouped Orthogonal Arrays and Their Applications

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 choices when there is no prior knowledge on how the input variables affect the response and the objective of experiments is global fitting. When there is some prior knowledge on the underlying true function of the system or what statistical models are more appropriate, a natural question is, are there more suitable designs than vanilla space-filling designs? In this article, we provide an answer for the cases where there are no interactions between the factors from disjoint groups of variables. In other words, we consider the design issue when the underlying functional form of the system or the statistical model to be used is additive where each component depends on one group of variables from a set of disjoint groups. For such cases, we recommend using {\em grouped orthogonal arrays.} Several construction methods are provided and many designs are tabulated for practical use. Compared with existing techniques in the literature, our construction methods can generate many more designs with flexible run sizes and better within-group projection properties for any prime power number of levels.

stat.ME↗

arXiv2024

Zarankiewicz numbers near the triple system threshold

For positive integers $m$ and $n$, the Zarankiewicz number $Z_{2,2}(m,n)$ can be defined as the maximum total degree of a linear hypergraph with $m$ vertices and $n$ edges. Guy determined $Z_{2,2}(m,n)$ for all $n \geq \binom{m}{2}/3+O(m)$. Here, we extend this by determining $Z_{2,2}(m,n)$ for all $n \geq \binom{m}{2}/3$ and, when $m$ is large, for all $n \geq \binom{m}{2}/6+O(m)$.

math.CO↗

arXiv2024

Exact values for unbalanced Zarankiewicz numbers

For positive integers $s$, $t$, $m$ and $n$, the Zarankiewicz number $Z_{s,t}(m,n)$ is defined to be the maximum number of edges in a bipartite graph with parts of sizes $m$ and $n$ that has no complete biparitite subgraph containing $s$ vertices in the part of size $m$ and $t$ vertices in the part of size $n$. A simple argument shows that, for each $t \geq 2$, $Z_{2,t}(m,n)=(t-1)\binom{m}{2}+n$ when $n \geq (t-1)\binom{m}{2}$. Here, for large $m$, we determine the exact value of $Z_{2,t}(m,n)$ in almost all of the remaining cases where $n=Θ(tm^2)$. We establish a new family of upper bounds on $Z_{2,t}(m,n)$ which complement a family already obtained by Roman. We then prove that the floor of the best of these bounds is almost always achieved. We also show that there are cases in which this floor cannot be achieved and others in which determining whether it is achieved is likely a very hard problem. Our results are proved by viewing the problem through the lens of linear hypergraphs and our constructions make use of existing results on edge decompositions of dense graphs.

math.CO↗

arXiv2023

New results on large sets of orthogonal arrays and orthogonal arrays

Orthogonal array and a large set of orthogonal arrays are important research objects in combinatorial design theory, and they are widely applied to statistics, computer science, coding theory and cryptography. In this paper, some new series of large sets of orthogonal arrays are given by direct construction, juxtaposition construction, Hadamard construction, finite field construction and difference matrix construction. Subsequently, many new infinite classes of orthogonal arrays are obtained by using these large sets of orthogonal arrays and Kronecker product.

math.CO↗

arXiv2020

Constructions of regular sparse anti-magic squares

Graph labeling is a well-known and intensively investigated problem in graph theory. Sparse anti-magic squares are useful in constructing vertex-magic labeling for graphs. For positive integers $n,d$ and $d<n$, an $n\times n$ array $A$ based on $\{0,1,\cdots,nd\}$ is called \emph{a sparse anti-magic square of order $n$ with density $d$}, denoted by SAMS$(n,d)$, if each element of $\{1,2,\cdots,nd\}$ occurs exactly one entry of $A$, and its row-sums, column-sums and two main diagonal sums constitute a set of $2n+2$ consecutive integers. An SAMS$(n,d)$ is called \emph{regular} if there are exactly $d$ positive entries in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares of order $n\equiv1,5\pmod 6$, and it is proved that for any $n\equiv1,5\pmod 6$, there exists a regular SAMS$(n,d)$ if and only if $2\leq d\leq n-1$.

math.CO↗

arXiv2019

Further Results on the Pseudo-$L_{g}(s)$ Association Scheme with $g\geq 3$, $s\geq g+2$

It is inevitable that the $L_{g}(s)$ association scheme with $g\geq 3, s\geq g+2$ is a pseudo-$L_{g}(s)$ association scheme. On the contrary, although $s^2$ treatments of the pseudo-$L_{g}(s)$ association scheme can form one $L_{g}(s)$ association scheme, it is not always an $L_{g}(s)$ association scheme. Mainly because the set of cardinality $s$, which contains two first-associates treatments of the pseudo-$L_{g}(s)$ association scheme, is non-unique. Whether the order $s$ of a Latin square $\mathbf{L}$ is a prime power or not, the paper proposes two new conditions in order to extend a $POL(s,w)$ containing $\mathbf{L}$. It has been known that a $POL(s,w)$ can be extended to a $POL(s,s-1)$ so long as Bruck's \cite{brh} condition $s\geq \frac{(s-1-w)^4-2(s-1-w)^3+2(s-1-w)^2+(s-1-w)}{2}$ is satisfied, Bruck's condition will be completely improved through utilizing six properties of the $L_{w+2}(s)$ association scheme in this paper. Several examples are given to elucidate the application of our results.

math.CO↗