Efficient Encoding of Data into Two-Dimensional Constrained Bit Patterns

Abstract

Two-dimensional constrained coding is a problem that is much more difficult than its one-dimensional counterpart. Indeed, in two dimensions, obtaining the answers to very natural questions becomes uncomputable. In particular, it is undecidable to determine if it is possible to fill the infinite plane with symbols in such a way that no forbidden pattern appears. Also, even when we know that such an infinite plane exists, it is uncomputable to determine the maximal rate at which payload data can be embedded into the selection of a valid infinite plane. Recently, Nakamura et al. presented a technique that efficiently performs the construction of a matrix of symbols that embeds payload data. Their technique is efficient in the sense that the construction takes time that is proportional to the area of the constructed matrix. Their technique is based on the offline elaboration of a collection of tiles, which is then used for the matrix construction. The collection-elaboration step is time consuming and it might even never terminate nor succeed. In this work, we extend their technique by generalizing their notion of tile. Our technique has the potential to achieve much higher data-embedding rates.