The existence of circular Florentine arrays

A κ × n circular Florentine array is an array of n distinct symbols in gk circular rows such that 1. (1) each row contains every symbol exactly once, and 2. (2) for any pair of distinct symbols (a, b) and for any integer m from 1 to n − 1 there is at most one row in which b occurs m steps to the right of a. For each positive integer n = 2, 3, 4,…, define Fc(n) to be the maximum number such that an Fc(n) × n circular Florentine array exists. From the main construction of this paper for a set of mutually orthogonal Latin squares (MOLS) having an additional property, and from the known results on the existence/nonexistence of such MOLS obtained by others, it is now possible to reduce the gap between the upper and lower bounds on Fc(n) for infinitely many additional values of n not previously covered. This is summarized in the table for all odd n up to 81.