  Motivated by dipolar-coupled artificial spin systems, we present a
theoretical study of the classical $J_1-J_2-J_3$ Ising antiferromagnet on the
kagome lattice. We establish the ground-state phase diagram of this model for
$J_1 > |J_2|, |J_3|$ based on exact results for the ground-state energies. When
all the couplings are antiferromagnetic, the model has three macroscopically
degenerate ground-state phases, and using tensor networks, we can calculate the
entropies of these phases and of their boundaries very accurately. In two
cases, the entropy appears to be a fraction of that of the triangular lattice
Ising antiferromagnet, and we provide analytical arguments to support this
observation. We also notice that, surprisingly enough, the dipolar ground state
is not a ground state of the truncated model, but of the model with smaller
$J_3$ interactions, an indication of a very strong competition between
low-energy states in this model.
