Folding, Tiling, and Multidimensional Coding

Abstract: Folding a sequence $S$ into a multidimensional box is a method that is used to construct multidimensional codes. The well known operation of folding is generalized in a way that the sequence $S$ can be folded into various shapes. The new definition of folding is based on lattice tiling and a direction in the $D$-dimensional grid. There are potentially $\frac{3^D-1}{2}$ different folding operations. Necessary and sufficient conditions that a lattice combined with a direction define a folding are given. The immediate and most impressive application is some new lower bounds on the number of dots in two-dimensional synchronization patterns. This can be also generalized for multidimensional synchronization patterns. We show how folding can be used to construct multidimensional error-correcting codes and to generate multidimensional pseudo-random arrays.