A construction of curves over finite fields (Q2717625)

\textit{G. van der Geer} and \textit{M. van der Vlugt} [Finite Fields Appl. 6, 327-341 (2000; Zbl 1021.11023)] produced curves with many rational points by considering Kummer covers of the form \(y^{q-1}=f(x)\), where \(f(x)\) is a rational function that assumes the value 1 at many elements of \({\mathbb F}_q\). The authors use this idea and consider Kummer covers of \({\mathbb P}^1_{{\mathbb F}_{q^n}}\) of the form \(y^m=g(x)/R(g(x))\), where \(m\) is a divisor of \(q^n-1\), \(g(x)\) is a polynomial, and \(R(g(x))\) is the remainder obtained when \(g(x)\) is divided by \(x^{q^n}-x\). By carefully choosing the polynomial \(g(x)\), the authors are able to obtain curves that set or tie records for the number of rational points for curves of that genus in many cases.