Embedding dimensions of matrices whose entries are indefinite distances in the pseudo-Euclidean space

A finite set of the Euclidean space is called an $s$-distance set provided the number of Euclidean distances in the set is $s$. Determining the largest possible $s$-distance set for the Euclidean space of a given dimension is challenging. This problem was solved only when dealing with small values of $s$ and dimensions. Lisoněk (1997) achieved the classification of the largest 2-distance sets for dimensions up to $7$, using computer assistance and graph representation theory. In this study, we consider a theory analogous to these results of Lisoněk for the pseudo-Euclidean space ${\mathbb{R}}^{p,q}$. We consider an $s$-indefinite-distance set in a pseudo-Euclidean space that uses the value $$\mathrm{\mid }\mathrm{\mid }x-y\mathrm{\mid }\mathrm{\mid }=(x_1-y_1{)}^2+\cdots +(x_p-y_p{)}^2-(x_{p+1}-y_{p+1}{)}^2-\cdots -(x_{p+q}-y_{p+q}{)}^2$$ instead of the Euclidean distance. We develop a representation theory for symmetric matrices in the context of $s$-indefinite-distance sets, which includes or improves the results of Euclidean $s$-distance sets with large $s$ values. Moreover, we classify the largest possible $2$-indefinite-distance sets for small dimensions.