  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 $N^{1-\epsilon}$ for
any $\epsilon>0$. The conjecture is that there is a constant $C>0$ such that
for any $n$-dimensional torus ${\mathbb T}^n={\mathbb R}^n/\Lambda$, where
$\Lambda$ is a lattice, the least volume unoriented $n/2$-dimensional surface
(using the Euclidean metric) representing nontrivial homology has volume at
least $C^n$ times the volume of the least volume $n/2$-dimensional hyperplane
representing nontrivial homology; in fact, it would suffice to have this result
for $\Lambda$ an integral lattice with the surface restricted to faces of a
cubulation by unit hypercubes. The main technical result is an estimate of
Rankin invariants\cite{rankin} 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\cite{qltc}). We also provide an short, alternative proof that the
shortest vector in the exterior power of a lattice may be
non-split\cite{coulangeon}.
