LOZENGE TILING CONSTRAINED CODES

While the field of one-dimensional constrained codesis mature, with theoretical as well as practical aspects of codeanddecoder-design being well-established, such a theoreticaltreatment of its two-dimensional (2D) counterpart is still unavailable.Research has been conducted on a few exemplar2D constraints, e.g., the hard triangle model, run-length limitedconstraints on the square lattice, and 2D checkerboardconstraints. Excluding these results, 2D constrained systemsremain largely uncharacterized mathematically, with only loosebounds of capacities present. In this paper we present a lozengeconstraint on a regular triangular lattice and derive Shannonnoiseless capacity bounds. To estimate capacity of lozenge tilingwe make use of the bijection between the counting of lozengetiling and the counting of boxed plane partitions

Asymptotic Capacity of Two-dimensional Channels with Checkerboard Constraints

A checkerboard constraint is a bounded measurable set S R , containing the origin. A binary labeling of the lattice satisfies the checkerboard constraint S if whenever t 2 Z is labeled 1, all of the other Z -lattice points in the translate t + S are labeled 0. Two-dimensional channels that only allow labelings of Z satisfying checkerboard constraints are studied. Let A (S) be the area of S, and let A (S) ! 1 mean that S retains its shape but is inflated in size in the form S, as ! 1. It is shown that for any open checkerboard constraint S, there exist positive reals K1 and K2 such that as A (S) ! 1, the channel capacity CS decays to zero at least as fast as (K1 log 2 A (S))=A (S) and at most as fast as (K2 log 2 A (S))=A (S). It is also shown that if S is an open convex and symmetric checkerboard constraint, then as A (S) ! 1, the capacity decays exactly at the rate 4(S)(log 2 A (S))=A (S), where (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 (S) = 1 for such constraints

Asymptotic Capacity of Two-Dimensional Channels with Checkerboard Constraints

One-dimensional channels satisfying run length constraints are important in magnetic recording applications and two-dimensional channels satisfying run length constraints have been considered in relation to optical recording applications (see the references in [1]). One-dimensional ¢¡¤£¦¥¨ § run length constraints require that in any binary sequence, there be at least ¡ and at most ¥ 0s between consecutive 1s. Two-dimensional run length constraints require that onedimensional run length constraints be satisfied both horizontally and vertically in a two-dimensional rectangular binary array. In addition to run length constraints, other types of constraints can be used to model certain two-dimensional channels. An example of a circularly symmetric two-dimensional constraint occurs by requiring that any point in the two-dimensional ©� � lattice be labeled 0 if it is within a prescribed distance from a lattice point with label 1. One could alternatively require that every 1 be surrounded by 0s falling in a given sized hexagon, square, or more generally any other shape of interest. In general, a large class of such two-dimensional constraints can be characterized by some bounded measurable twodimensional set � , and the requirement that for every 1 stored in the plane, it must at least be surrounded by a set of 0s arranged in the shape of �. Such a code is said to satisfy the constraint �. These constraints are known as checkerboard constraints [3]. For a convex symmetric checkerboard constraints � , we deter-mine the rate at which the capacity goes to zero, as a function of the