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Folding a sequence $S$ into a multidimensional box is a well-known methodwhich is used as a multidimensional coding technique. The operation of foldingis generalized in a way that the sequence $S$ can be folded into various shapesand not just a box. The new definition of folding is based on a lattice tilingfor the given shape $\cS$ and a direction in the $D$-dimensional integer grid.Necessary and sufficient conditions that a lattice tiling for $\cS$ combinedwith a direction define a folding of a sequence into $\cS$ are derived. Theimmediate and most impressive application is some new lower bounds on thenumber of dots in two-dimensional synchronization patterns. This can be alsogeneralized for multidimensional synchronization patterns. The technique andits application for two-dimensional synchronization patterns, raise someinteresting problems in discrete geometry. We will also discuss these problems.It is also shown how folding can be used to construct multidimensionalerror-correcting codes. Finally, by using the new definition of folding,multidimensional pseudo-random arrays with various shapes are generated.

Sequence Folding, Lattice Tiling, and Multidimensional Coding