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 is an element of F-nxn is defined as l(1)l(2) ... l(n), where l(j) is an element of {A, S, 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 F = F-2, we initiate a study of the case F = F-3. Moreover, epr-sequences over finite fields are shown to have connections to Ramsey theory and coding theory.