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Motivated by dipolar-coupled artificial spin systems, we present atheoretical study of the classical $J_1-J_2-J_3$ Ising antiferromagnet on thekagome 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. Whenall the couplings are antiferromagnetic, the model has three macroscopicallydegenerate ground-state phases, and using tensor networks, we can calculate theentropies of these phases and of their boundaries very accurately. In twocases, the entropy appears to be a fraction of that of the triangular latticeIsing antiferromagnet, and we provide analytical arguments to support thisobservation. We also notice that, surprisingly enough, the dipolar ground stateis not a ground state of the truncated model, but of the model with smaller$J_3$ interactions, an indication of a very strong competition betweenlow-energy states in this model.

Partial lifting of degeneracy in the $J_1-J_2-J_3$ Ising antiferromagnet on the kagome lattice