Further Results on the Pseudo-$L_{g}(s)$ Association Scheme with $g\geq 3$, $s\geq g+2$

TL;DR

This paper improves conditions for extending partial orthogonal Latin squares using properties of specific association schemes, providing new criteria and examples for such extensions.

Contribution

It introduces two new conditions for extending partial orthogonal Latin squares by leveraging properties of $L_{g}(s)$ association schemes, refining Bruck's existing condition.

Findings

Improved bounds for extending partial Latin squares.

New criteria based on $L_{g}(s)$ association schemes.

Examples illustrating the application of the new conditions.

Abstract

It is inevitable that the $L_{g}(s)$ association scheme with $g\geq 3, s\geq g+2$ is a pseudo-$L_{g}(s)$ association scheme. On the contrary, although $s^2$ treatments of the pseudo-$L_{g}(s)$ association scheme can form one $L_{g}(s)$ association scheme, it is not always an $L_{g}(s)$ association scheme. Mainly because the set of cardinality $s$, which contains two first-associates treatments of the pseudo-$L_{g}(s)$ association scheme, is non-unique. Whether the order $s$ of a Latin square $\mathbf{L}$ is a prime power or not, the paper proposes two new conditions in order to extend a $POL(s,w)$ containing $\mathbf{L}$. It has been known that a $POL(s,w)$ can be extended to a $POL(s,s-1)$ so long as Bruck's \cite{brh} condition $s\geq \frac{(s-1-w)^4-2(s-1-w)^3+2(s-1-w)^2+(s-1-w)}{2}$ is satisfied, Bruck's condition will be completely improved through utilizing six properties of the $L_{w+2}(s)$ association scheme in this paper. Several examples are given to elucidate the application of our results.