Diagonal cellular factor pair Latin squares

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 \(p^n/n\).