  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.
