Asymptotic capacity of two-dimensional channels with checkerboard constraints

A checkerboard constraint is a bounded measurable set S⊂R², containing the origin. A binary labeling of the Z² lattice satisfies the checkerboard constraint S if whenever t∈Z² is labeled 1, all of the other Z²-lattice points in the translate t+S are labeled 0. Two-dimensional channels that only allow labelings of Z² satisfying checkerboard constraints are studied. Let A(S) be the area of S, and let A(S)→∞ mean that S retains its shape but is inflated in size in the form αS, as α→∞. It is shown that for any open checkerboard constraint S, there exist positive reals K₁ and K₂ such that as A(S)→∞, the channel capacity C_S decays to zero at least as fast as (K₁log₂A(S))/A(S) and at most as fast as (K₂log₂A(S))/A(S). It is also shown that if S is an open convex and symmetric checkerboard constraint, then as A(S)→∞, the capacity decays exactly at the rate 4δ(S)(log₂A(S))/A(S), where δ(S) is the packing density of the set S. An implication is that the capacity of such checkerboard constrained channels is asymptotically determined only by the areas of the constraint and the smallest (possibly degenerate) hexagon that can be circumscribed about the constraint. In particular, this establishes that channels with square, diamond, or hexagonal checkerboard constraints all asymptotically have the same capacity, since δ(S)=1 for such constraints.