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

Abstract

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 ${\ell }_j\in \{\mathtt{A},\mathtt{S},\mathtt{N}\}$ 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.