We construct toric codes on various high-dimensional manifolds. Assuming a conjecture in geometry we find families of quantum CSS stabilizer codes on N qubits with logarithmic weight stabilizers and distance N 1− for any > 0. The conjecture is that there is a constant C > 0 such that for any n-dimensional torus T n = R n /Λ, where Λ is a lattice, the least volume unori-ented n/2-dimensional cycle (using the Euclidean metric) representing nontrivial homology has volume at least C n times the

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... me of the least volume n/2-dimensional hyperplane representing nontrivial homology; in fact, it would suffice to have this result for Λ an integral lattice with the cycle restricted to faces of a cubulation by unit hypercubes. The main technical result is an estimate of Rankin invariants[24] for certain random lattices, showing that in a certain sense they are optimal. Additionally, we construct codes with square-root distance, logarithmic weight stabilizers, and inverse polylogarithmic soundness factor (considered as quantum locally testable codes[1]). We also provide an short, alternative proof that the shortest vector in the exterior power of a lattice may be non-split[8].