On an isoperimetric problem for Hamming graphs

We consider the vertex-isoperimetric problem for the product of complete graphs. This was posed to us by A.R. Urbanke who wished to apply it in error-correcting coding theory. The problem does not have nested solutions, so the known combinatorial methods are ineffectual. However, passing to the continuous limit gives a variational problem, interesting in its own right, which we solve exactly. Its solution gives a lower bound on the original combinatorial problem which is asymptotically best possible and even sharp in many cases.