  We propose a family of local CSS stabilizer codes as possible candidates for
self-correcting quantum memories in 3D. The construction is inspired by the
classical Ising model on a Sierpinski carpet fractal, which acts as a classical
self-correcting memory. Our models are naturally defined on fractal subsets of
a 4D hypercubic lattice with Hausdorff dimension less than 3. Though this does
not imply that these models can be realised with local interactions in 3D
Euclidean space, we also discuss this possibility. The X and Z sectors of the
code are dual to one another, and we show that there exists a finite
temperature phase transition associated with each of these sectors, providing
evidence that the system may robustly store quantum information at finite
temperature.
