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 ∈ F n×n is defined as ℓ 1 ℓ 2 • • • ℓ n , where ℓ j ∈ {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.