Quadratic forms of codimension 2 over certain finite fields of even characteristic

Abstract

Let ${\mathbb F}_q$ be a finite field of characteristic 2, not containing ${\mathbb F}_4$ . Let k???2 be an even integer. We give a full classification of quadratic forms over ${\mathbb F}_{q^k}$ of codimension 2 provided that certain three coefficients are from ${\mathbb F}_4$ . We apply this to the classification of maximal and minimal curves over finite fields.