An order-m Latin square is a factor pair Latin square if, for each ordered pair (a, b) of positive integers satisfying \(ab=m\), there is no repetition of symbols in any \(a\times b\) tiling region. If p is prime and n is a natural number, we introduce a new characterization of order-\(p^n\) linear factor pair Latin squares. We use this characterization to show that the Latin squares constructed by Mariot et al. (Des Codes Cryptogr 88:391-411, 2020) corresponding to \({{\mathbb {F}}}_q\)- linear bipermutive cellular automata are linear factor pair Latin squares when q is prime. These linear factor pair Latin squares are called cellular factor pair Latin squares. We then apply the results and methods of Mariot et al. (2020) to construct maximally sized families of pairwise mutually orthogonal cellular factor pair Latin squares of order \(p^n\), and likewise for diagonal cellular factor pair Latin squares of order \(p^n\). The size of each of these two families is asymptotic to \(p^n/n\).

阶m拉丁方是因子对拉丁方，如果对于满足\(ab=m\)的正整数的每个有序对 ( a ,  b ) ，在任何\(a\times b\ )平铺区域。如果p是素数，n是自然数，我们引入一个新的阶表征—— \(p^n\)线性因子对拉丁方。我们使用此特征来表明 Mariot 等人构建的拉丁方。(Des Codes Cryptogr 88:391-411, 2020) 对应于\({{\mathbb {F}}}_q\) - 线性双排列元胞自动机是线性因子对拉丁方当q是质数。这些线性因子对拉丁方称为细胞因子对拉丁方。然后我们应用 Mariot 等人的结果和方法。(2020) 构建成对相互正交的元胞因子对拉丁方的最大尺寸族\(p^n\)，同样对于对角元胞因子对拉丁方\(p^n\)。这两个系列中每个系列的大小都渐近于\(p^n/n\)。