Further Results on the Pseudo-$L_g(s)$ Association Scheme with $g\ge 3$, $s\ge g+2$

Abstract

It is inevitable that the $L_g(s)$ association scheme with $g\ge 3,s\ge g+2$ 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 $s\ge \frac{(s-1-w{)}^4-2(s-1-w{)}^3+2(s-1-w{)}^2+(s-1-w)}{2}$ 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.

52 pages