The Enumeration of Cyclic MNOLS

TL;DR

This paper investigates cyclic MNOLS, determining the maximum number for orders up to 18 and providing comprehensive enumeration, thereby disproving a previous conjecture about their maximum size.

Contribution

It establishes the maximum size of cyclic MNOLS for orders up to 18 and fully enumerates their configurations, challenging existing conjectures.

Findings

Maximum $ ext{MNOLS}$ sets identified for $n \,\leq\, 18$

Complete enumeration of cyclic $ ext{MNOLS}$ for these orders

Disproof of the conjecture relating maximum size to $\lceil n/4 \rceil + 1$

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 $\mu$ such that there exists a set of $\mu$ cyclic $\text{MNOLS}$ of order $n$ for $n \leq 18$, as well as providing a full enumeration of sets and lists of $\mu$ 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 $\mu$ for which a set of $\mu$ cyclic $\text{MNOLS}$ of order $n$ exists is $\lceil n/4\rceil +1$.