The Enumeration of Cyclic MNOLS

Abstract

In this paper we study collections of mutually nearly orthogonal Latin squares ($\text{MNOLS}$), which come from a modification of the orthogonal condition for mutually orthogonal Latin squares. In particular, we find the maximum $μ$ such that there exists a set of $μ$ cyclic $\text{MNOLS}$ of order $n$ for $n \leq 18$, as well as providing a full enumeration of sets and lists of $μ$ cyclic $\text{MNOLS}$ of order $n$ under a variety of equivalences with $n \leq 18$. This resolves in the negative a conjecture that proposed the maximum $μ$ for which a set of $μ$ cyclic $\text{MNOLS}$ of order $n$ exists is $\lceil n/4\rceil +1$.