论文标题
拉丁高管和细胞自动机
Latin Hypercubes and Cellular Automata
论文作者
论文摘要
拉丁正方形和高管是统计，密码学和编码理论的多种应用的组合设计。在本文中，我们将基于双层蜂窝自动机（CA）的拉丁正方形的结构推广到尺寸的拉丁高管$ k> 2 $。特别是，我们证明了线性双臂CA（LBCA）产生的拉丁尺寸$ k> 2 $的拉丁高管由可逆的toeplitz矩阵的序列定义，具有部分重叠系数的序列，可以由特定类型的常规DE BRUIJN图描述，这些de bruijn图由确定性功能的支持。此外，我们得出了LBCA生成的$ K $维拉丁超级立方体的数量，这是通过计算此de bruijn图上的长度$ k-3 $的数量。
Latin squares and hypercubes are combinatorial designs with several applications in statistics, cryptography and coding theory. In this paper, we generalize a construction of Latin squares based on bipermutive cellular automata (CA) to the case of Latin hypercubes of dimension $k>2$. In particular, we prove that linear bipermutive CA (LBCA) yielding Latin hypercubes of dimension $k>2$ are defined by sequences of invertible Toeplitz matrices with partially overlapping coefficients, which can be described by a specific kind of regular de Bruijn graph induced by the support of the determinant function. Further, we derive the number of $k$-dimensional Latin hypercubes generated by LBCA by counting the number of paths of length $k-3$ on this de Bruijn graph.
