F-squares over cellular automata.

Abstract

F-squares are a class of arrays in which each element occurs the same number of times in every row and column; they play a significant role in experimental design. This paper examines the conditions under which types of F-squares can be generated by cellular automata (CAs). The approach is based on analysing the row and column sums of the arrays produced by CAs over 𝔽2, recognising that equal sums are a defining characteristic of F-squares. Particular attention is given to pairs of columns whose binary representations differ only in their final bit, and to pairs of rows whose binary representations differ only in their first bit, with the aim of identifying local rules that ensure equal column sums and equal row sums. Two families of local rules are identified that satisfy these conditions: bipermutive local rules and constant local rules. These results reveal that every F-square generated by a CA over 𝔽2 is either a Latin square or a trivial F-square.