A comparision of the number of rational places of certain function fields to the Hasse-Weil bounds (Q1879276)

This paper studies the number \(Z_l\) of \(E_l\)-rational points on \(x^m+y^n=1\) for extension fields \(E_l=F_{2^{kl}}\) of \(F_{2^k}\) with \(k=\Phi(N)/2\) and \(N=\text{lcm}(n,m)\) and on the projective non-singular model of the curve, denoted by \(N_l\). The study is motivated by applications in coding theory where a large number of points is desirable and they demonstrate that in the cases they consider the number of points is close to the upper Hasse-Weil bound for many cases. The exact number of points is easy to state if \(-1\in \langle 2 \rangle\) in \(\mathbb Z_N\); in this paper the authors consider the remaining cases and give explicit formulae for the case that \(N=pq\). This study is split further into 3 cases according to the residues of \(p\) and \(q\) modulo \(4\). The proofs use exponential sums and the number of points (depending on the choice of \(m\) and \(n\) in \(\{p, q, pq\}\)) is stated in terms of the solution of a Diophantine equation. The latter can be solved efficiently and this was used by the same authors in [IEEE Trans. Inf. 45, No. 4, 1244--1249 (1999; Zbl 0959.94027)] to obtain the weight distribution of binary codes. For each of the 3 cases they show that there are infinitely many extension degrees \(l\) such that \(N_l\) is arbitrarily close to the Hasse-Weil bound.