Combinatorial properties of the enhanced principal rank characteristic sequence over finite fields

The enhanced principal rank characteristic sequence (epr-sequence) of a symmetric matrix $B \in \mathbb{F}^{n \times n}$ is defined as $\ell_1 \ell_2 \cdots \ell_n$, where $\ell_j \in \{\tt{A}, \tt{S}, \tt{N}\}$ according to whether all, some but not all, or none of the principal minors of order $j$ of $B$ are nonzero. Building upon the second author's recent classification of the epr-sequences of symmetric matrices over the field $\mathbb{F}=\mathbb{F}_2$, we initiate a study of the case $\mathbb{F}=\mathbb{F}_3$. Moreover, epr-sequences over finite fields are shown to have connections to Ramsey theory and coding theory.