Quadratic forms of codimension 2 over finite fields containing F4 and Artin–Schreier type curves

Let Fq be a finite field containing F4. Let k⩾2 be an integer. We give a full classification of quadratic forms over Fqk of codimension 2 provided that certain three coefficients are from F4. As an application of this we obtain new results on the classification of maximal and minimal curves over Fqk. We also give some nonexistence results on certain systems of equations over Fqk.