In this paper, we enumerate superspecial hyperelliptic curves of genus 4 over finite fields F_q for small q. This complements our preceding results in the non-hyperelliptic case. We give a feasible algorithm to enumerate superspecial hyperelliptic curves of genus g over F_q in the case that q and 2g + 2 are coprime and q > 2g + 1. We executed the algorithm for (g,q) = (4, 11~2), (4,13~2), (4,17~2) and (4,19) with our implementation on a computer algebra system Magma. Moreover, we found many maximal hyperelliptic curves and some minimal hyperelliptic curves over F_q from among enumerated superspecial curves.
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