  A Richelot isogeny between Jacobian varieties is an isogeny whose kernel is
included in the $2$-torsion subgroup of the domain. A Richelot isogeny whose
codomain is the product of two or more principally polarized abelian varieties
is called a decomposed Richelot isogeny.
  In this paper, we develop some explicit arithmetic on curves of genus $3$,
including algorithms to compute the codomain of a decomposed Richelot isogeny.
As solutions to compute the domain of a decomposed Richelot isogeny, explicit
formulae of defining equations for Howe curves of genus $3$ are also given.
Using the formulae, we shall construct an algorithm with complexity
$\tilde{O}(p^3)$ (resp. $\tilde{O}(p^4)$) to enumerate all hyperelliptic (resp.
non-hyperelliptic) superspecial Howe curves of genus $3$.
