Latin Hypercubes and Cellular Automata

On the number of frequency hypercubes $F^n(4;2,2)$

A frequency $n$-cube $F^n(4;2,2)$ is an $n$-dimensional $4$-by-...-by-$4$ array filled by $0$s and $1$s such that each line contains exactly two $1$s. We classify the frequency $4$-cubes $F^4(4;2,2)$, find a testing set of size $25$ for $F^3(4;2,2)$, and derive an upper bound on the number of $F^n(4;2,2)$. Additionally, for any $n$ greater than $2$, we construct an $F^n(4;2,2)$ that cannot be refined to a latin hypercube, while each of its sub-$F^{n-1}(4;2,2)$ can. Keywords: frequency hypercube, frequency square, latin hypercube, testing set, MDS code

Discrepancy Bounds for a Class of Negatively Dependent Random Points Including Latin Hypercube Samples

We introduce a class of $γ$-negatively dependent random samples. We prove that this class includes, apart from Monte Carlo samples, in particular Latin hypercube samples and Latin hypercube samples padded by Monte Carlo. For a $γ$-negatively dependent $N$-point sample in dimension $d$ we provide probabilistic upper bounds for its star discrepancy with explicitly stated dependence on $N$, $d$, and $γ$. These bounds generalize the probabilistic bounds for Monte Carlo samples from [Heinrich et al., Acta Arith. 96 (2001), 279--302] and [C.~Aistleitner, J.~Complexity 27 (2011), 531--540], and they …

Mutually Orthogonal Latin Squares based on Cellular Automata

We investigate sets of Mutually Orthogonal Latin Squares (MOLS) generated by Cellular Automata (CA) over finite fields. After introducing how a CA defined by a bipermutive local rule of diameter $d$ over an alphabet of $q$ elements generates a Latin square of order $q^{d-1}$, we study the conditions under which two CA generate a pair of orthogonal Latin squares. In particular, we prove that the Latin squares induced by two Linear Bipermutive CA (LBCA) over the finite field $\mathbb{F}_q$ are orthogonal if and only if the polynomials associated to their local rules are relatively prime. Next, w…

On Negative Dependence Properties of Latin Hypercube Samples and Scrambled Nets

We study the notion of $γ$-negative dependence of random variables. This notion is a relaxation of the notion of negative orthant dependence (which corresponds to $1$-negative dependence), but nevertheless it still ensures concentration of measure and allows to use large deviation bounds of Chernoff-Hoeffding- or Bernstein-type. We study random variables based on random points $P$. These random variables appear naturally in the analysis of the discrepancy of $P$ or, equivalently, of a suitable worst-case integration error of the quasi-Monte Carlo cubature that uses the points in $P$ as integra…

用线性元胞自动机构造正交拉丁方

Constructing Orthogonal Latin Squares from Linear Cellular Automata

我们进行调查的组合设计所产生的元胞自动机（CA），特别侧重于正交拉丁广场和正交阵列。动机是加密性质的。事实上，我们考虑的问题，采用CA通过正交拉丁方来定义门限秘密共享方案。我们首先展示了如何通过双置换CA生成拉丁方。然后，使用的西尔维斯特矩阵的基础上的一个特征，我们证明了两个线性CA诱导一对正交拉丁方当且仅当多项式相关联的本地规则是相对素。

自正交元胞自动机

Self-Orthogonal Cellular Automata

本文研究双置换元胞自动机生成自正交拉丁方的条件，枚举小直径规则并给出线性CA自正交性的完整理论刻画，证明二元情形下不可约性为充分条件，并将判定简化为局部规则中心系数奇偶性计算。