Folding, Tiling, and Multidimensional Coding

TL;DR

This paper introduces a generalized folding method based on lattice tiling and directions in multidimensional grids, enabling new code constructions and bounds for synchronization patterns and error correction.

Contribution

It generalizes the folding operation to various shapes using lattice tiling and directions, providing new theoretical bounds and applications in multidimensional coding.

Findings

New lower bounds on dots in 2D synchronization patterns

Generalized folding operation for various shapes

Applications in multidimensional error-correcting codes

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.