Let \(C\) be an algebraic curve defined over a finite field \(\mathbb{F}_ q\) of \(q\) elements whose function field \(\mathbb{F}_ q(C)\) is a finite extension of the pure transcendental field \(\mathbb{F}_ q(x)\). Let \(R(x)\in \mathbb{F}_ q(C)\) such that \(R(x) \not\equiv h(x)^ p -h(x)\), for any \(h\in \overline {\mathbb{F}}_ q(C)\). Denote by \(\mathbb{F}_ p\) the finite field of \(p\) elements and by \(\mathbb{F}_{q^ m}\) the extension of \(\mathbb{F}_ q\) of degree \(m\). Let \(\sigma: \mathbb{F}_{q^ m} \mapsto \mathbb{F}_ p\) be the relative trace map. The authors study the exponential sum \(\Psi_ m (R,C)= \sum_ P \psi (\sigma (R(P)))\), where \(\psi(t)= \exp (2\pi it/p)\) and \(P\) runs through the \(\mathbb{F}_{q^ m}\)-rational points of \(C\) which are not poles of \(R(x)\). Denote by \((R)_ \infty= \sum_{i=1}^ t d_ i P_ i\) the divisor of poles of \(R(x)\) in \(C\). Then the Bombieri-Weil bound [\textit{E. Bombieri}, Am. J. Math. 88, 71-105 (1966; Zbl 0171.415)] asserts that \[ | \Psi_ m (R,C)|\leq (2g-2 +t+ \deg(R)_ \infty) q^{m/2}, \] where \(g\) denotes the genus of \(C\). Moreover, this inequality is the best possible if \(\text{gcd} (d_ i,p) =1\), for every \(i=1, \dots, t\). The authors improve this result by using \textit{J. P. Serre's} [C. R. Acad. Sci., Paris, Sér. I 296, 397-402 (1983; Zbl 0538.14015)] sharpening of Weil's bound [\textit{A. Weil}, Courbes algébriques et variétés abéliennes (Hermann, Paris, 1971; Zbl 0208.492)] for the number of rational points of an algebraic curve over a finite field. More precisely, if \(N\) denotes the number of \(\mathbb{F}_ q\)- rational points of \(C\), then \[ | N- (q+1)| \leq g[2q^{1/2}]. \] Their result is the following: Let \(R(x)\) be a non-constant rational function on \(C\) and suppose that \(\text{char} (\mathbb{F}_ q) =2\), \(R(x)\not\equiv H(x)^ 2+ H(x)+ \alpha\), where \(H\in \mathbb{F}_ q (C)\) and \(\alpha\in \mathbb{F}_ q\). Thus \(C'\): \(y^ 2+y= R(x)\) is an Artin-Schreier covering of degree two of \(C\) and \(| \sum_ P (-1)^{\sigma (R(P))}| \leq (g'-g) [2\sqrt{q^ m}]\), where \(g'\) denotes the genus of \(C'\). Afterwards they apply this result to improve previous results on the minimum distance of the dual code of the Goppa code with polynomial \(G(x)\) with coefficients in \(\mathbb{F}_ q\), where \(q= 2^ m\) [\textit{C. J. Moreno} and \textit{O. Moreno}, Proc. Am. Math. Soc. 111, 523-531 (1991; Zbl 0716.94010)]. More specifically, this minimum distance is at least \[ 2^{m-1} -(k-1)/2- ((\deg G-2+s) [2^{m/2+1}]) /4. \] They also obtain that the minimum distance of the dual of the primitive error correcting BCH code of length \(2^ m -1\) has minimum distance at least \(2^{m-1}- (t-1) [2^{m/2+1}] /2\).