Some enumeration results on trace forms of low radical over finite fields of characteristic two

Abstract

We introduce some enumeration results for quadratic forms written as trace forms with very low dimension of radical for different types of degrees of extension over F2. This work shows a different approach to extend earlier enumeration results, for instance, when the degree of extension is product of two odd primes. We also determine the possible pairs of invariants for some trace forms and use those results to construct some new maximal Artin-Schreier curves with co-dimension 4 and 6 when the degrees of extension are divisible by 8 but not by 16.