turkmath.org

Given two Latin squares $L_1$ and $L_2$ of the same order, the hamming distance between $L_1$ and $L_2$ gives the number of corresponding cells containing distinct symbols. If we think of Latin squares as sets of ordered triples, this is given by |$L_1$ \ $L_2$|. Given a specific Latin square $L_1$, we may wish to know a Latin square $L_2$ which is closest to it; i.e. for which the Hamming distance is minimized. Equivalently, we may ask for the size of the smallest Latin trade within a given Latin square. It is known that the back circulant Latin square of order $n$ (the operation table for the integers modulo n) has Hamming distance at least $e \log n + 2$ to any other Latin square. We explore whether the back circulant Latin square is the loneliest of all Latin squares; i.e. has greatest minimum Hamming distance to any other Latin square. This is a joint work with R. Ramadurai.