Exponential sums, codes, and Hermitian forms

We defined the quadratic Hermitian forms on a space of even dimension over a finite field; they are constructed from classical Hermitian forms. We give the exact value of exponential sums associated to the quadratic Hermitian forms. On the other hand, we define with the space of quadratic Hermitian forms a new family of subcodes of the Reed-Muller code of order two, and we give the value of the parameters of these codes and of the associated trace codes.