The enumeration of cyclic mutually nearly orthogonal Latin squares

Abstract

In this paper, we study collections of mutually nearly orthogonal Latin squares (MNOLS), which come from a modification of the orthogonality condition for mutually orthogonal Latin squares. In particular, we find the maximum mu such that there exists a set of mu cyclic MNOLS of order n for n <= 18, as well as providing a full enumeration of sets and lists of mu cyclic MNOLS of order n under a variety of equivalences with n <= 18. This resolves in the negative a conjecture that proposed that the maximum mu for which a set of mu cyclic MNOLS of order n exists is left ceiling n/4 right ceiling +1.