An improvement to the Hasse-Weil bound and applications to character sums, cryptography and coding

In this paper, we focus on the most interesting case for applications, namely p=2. We show that the Hasse-Weil bound for this special family of curves can be improved if q=2n with odd n⩾3 which is the same case where Serre [10] improved the Hasse-Weil bound. However, our improvement is greater than Serre's and Moreno-Morenao's improvements for this special family of curves. Furthermore, our improvement works for p=2 compared with the requirement of large p by Rojas-Leon and Wan. In addition, our improvement finds interesting applications to character sums, cryptography and coding theory. The key idea behind is that this curve has the Hasse-Witt invariant 0 and we show that the Hasse-Weil bound can be improved for any curves with the Hasse-Witt invariant 0. The main tool used in our proof involves Newton polygon and some results in algebraic geometry.