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Sequence Folding, Lattice Tiling, and Multidimensional Coding

IEEE TRANSACTIONS ON INFORMATION THEORY 2011年第7期57卷 4383-4400页

作者： Etzion, Tuvi Technion Israel Inst Technol Dept Comp Sci IL-32000 Haifa Israel

Folding a sequence into a multidimensional box is a well-known method which is used as a multidimensional coding technique. The operation of folding is generalized in a way that the sequence can be folded into various shapes and not just a box. The novel definition of folding is based on a lattice tiling for the given shape and a direction in the D-dimensional integer grid. Necessary and sufficient conditions that a lattice tiling for combined with a direction define a folding of a sequence into are derived. The immediate and most impressive applications are some new lower bounds on the number of dots in two-dimensional synchronization patterns. Asymptotically optimal such patterns were known only for rectangular shapes. We show asymptotically optimal such patterns for a large family of hexagons. This is also generalized for multidimensional synchronization patterns. The best known patterns, in terms of dots, for circles and other polygons are also given. The technique and its application for two-dimensional synchronization patterns, raises some interesting problems in discrete geometry. We will also discuss these problems. It is also shown how folding can be used to construct multidimensional error-correcting codes. Finally, by using the new definition of folding, new types of multidimensional pseudo-random arrays with various shapes are generated.

关键词： Distinct difference configuration folding lattice tiling pseudo-random array two-burst-correcting code

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