Algorithms are given to enumerate superspecial hyperelliptic curves of genus g over F_q (when q and 2g+2 coprime, q>2g+1) and to compute their automorphism groups, with concrete results for g=4 over small F_p and F_p^2.
and Harashita, S.: Enumerating superspecial curves of genus 4 over prime fields , arXiv: 1702.05313 [math.AG], 2017
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Algorithmic study of superspecial hyperelliptic curves over finite fields
Algorithms are given to enumerate superspecial hyperelliptic curves of genus g over F_q (when q and 2g+2 coprime, q>2g+1) and to compute their automorphism groups, with concrete results for g=4 over small F_p and F_p^2.