Capacity of Hexagonal Checkerboard Codes

CHECKERBOARD codes are two-dimensional binary codes that are designed to satisfy specific constraints [1], [2]. An example is the two-dimensional rectangular binary arrays that satisfy the (d, k) run-length constraint— there are at most k 0’s and the number of 0’s between any neighboring 1’s is at least d in each row and column [3]–[7]. It may naturally arise from data storage on a surface [8], [9]. An example of the two dimensional (1, ∞) run-length constraint is shown in Fig. 1. In practice, the distance between two data recorded points on a recording device should be no less than a given threshold due to the physical and fidelity constraints. It is of interest to study the checkerboard code on the hexagonal lattice, because it is known that the lattice arrangement of circles with the highest density in two dimensional Euclidean space is the hexagonal packing arrangement, in which the centers of the circles are arranged on the hexagonal lattice [10]. Specifically, we study the capacity of the checkerboard code shown in Fig. 2, where only 0’s can be arranged in the six neighbors of any 1, while both 0 and 1 can be arranged in the six neighbors of any 0 (which is referred to as the hexagonal constraint).