Hyperelliptic curves of genus three over finite fields of even characteristic (Q1827586)

The interest on hyperelliptic curves over a finite field of characteristic 2 of genus 2 or 3 was renewed after \textit{N. Koblitz}'s paper [J. Cryptology 1, 139--150 (1989; Zbl 0674.94010)] and \textit{L. M. Adleman}, \textit{J. DeMarrais} and \textit{M. Huang}'s paper [Algorithmic number theory, ANTS-I, 1994, Lect. Notes Comput. Sci. 877, 28--40 (1994; Zbl 0829.11068)], where connections with cryptography matters were established. Such curves of genus 2 were classified by \textit{G. Cardona}, \textit{E. Nart} and \textit{J. Pujolàs} [http:\slash\slash www.arxiv.org/math.NT/0210105]. In this paper, the genus 3 case is studied. By using a similar method as in the aforementioned paper, the authors succeed in classifying the curves in this case via rational models with a given structure of its ramification divisor.