  It is well-known that the second-order cone can be outer-approximated to an
arbitrary accuracy $\epsilon$ by a polyhedral cone of compact size defined by
irrational data. In this paper, we propose two rational polyhedral
outer-approximations of compact size retaining the same guaranteed accuracy
$\epsilon$. The first outer-approximation has the same size as the optimal but
irrational outer-approximation from the literature. In this case,we provide a
practical approach to obtain such an approximation defined by the smallest
integer coefficients possible, which requires solving a few, small-size integer
quadratic programs. The second outer-approximation has a size larger than the
optimal irrational outer-approximation by a linear additive factor in the
dimension of the second-order cone. However, in this case, the construction is
explicit, and it is possible to derive an upper bound on the largest
coefficient, which is sublinear in $\epsilon$ and logarithmic in the dimension.
We also propose a third outer-approximation, which yields the best possible
approximation accuracy given an upper bound on the size of its coefficients.
Finally, we discuss two theoretical applications in which having a rational
polyhedral outer-approximation is crucial, and run some experiments which
explore the benefits of the formulations proposed in this paper from a
computational perspective.
