Factor of i.i.d. with polynomial tails for the long-range Ising model

We prove that the very-high temperature long-range Ising model on ${\mathbf{Z}}^d$ is a factor of an i.i.d. process, and provide quantitative information on the coding radius. We do not assume that the model is ferromagnetic, but require the interactions to decay faster than $1\mathrm{/}{n}^{2d}$. While previous constructions were known, the novelty of our approach is that we obtain an explicit polynomial upper bound on the tail of the coding radius. In particular, our construction yields a factor map with finite expected coding volume. This answers a question of a recent preprint by Chazottes, Gallo and Takahashi, see arXiv:2602.14618.