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Lozenge tiling constrained codes

Vasić Bane (University of Arizona, Department of Electrical and Computer Engineering, Tucson, USA)
Krishnan Anantha Raman (Western Digital Corporation, Irvine, USA)

While the field of one-dimensional constrained codes is mature, with theoretical as well as practical aspects of code- and decoder-design being well-established, such a theoretical treatment of its two-dimensional (2D) counterpart is still unavailable. Research has been conducted on a few exemplar 2D constraints, e.g., the hard triangle model, run-length limited constraints on the square lattice, and 2D checkerboard constraints. Excluding these results, 2D constrained systems remain largely uncharacterized mathematically, with only loose bounds of capacities present. In this paper we present a lozenge constraint on a regular triangular lattice and derive Shannon noiseless capacity bounds. To estimate capacity of lozenge tiling we make use of the bijection between the counting of lozenge tiling and the counting of boxed plane partitions.

Keywords: 2D constrained codes, 2D constraints, lozenge tiling, colored tiling