Sequence Folding, Lattice Tiling, and Multidimensional Coding

Abstract

Folding a sequence S 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 S can be folded into various shapes and not just a box. The new definition of folding is based on a lattice tiling for the given shape S and a direction in the Ddimensional integer grid. Necessary and sufficient conditions that a lattice tiling for S combined with a direction define a folding of a sequence into S are derived. 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. The technique and its application for two-dimensional synchronization patterns, raise 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, multidimensional pseudorandom arrays with various shapes are generated.

Sidon sequences and doubly periodic two-dimensional synchronization patterns

“…In fact it should be said that is very likely that most if not all Costas arrays are known since they derived from the known constructions [25]. In what follows we will throw more evidence for the difficulty to produce new doubly periodic S-DDCs with many dots, different from those constructed by folding [4].…”

“…Given a tiling (T , S) and a grid point (i 1 , i 2 ) we denote by c(i 1 , i 2 ) the center of the copy of S, S ′ , for which (i 1 , i 2 ) ∈ S ′ . We will also assume that the origin is a center of a copy of S. The first lemma given in [4] can be easily verified.…”

“…• In each two copies of S in the tiling, the positions of the dots are the same. Doubly periodic DDCs and in particular Costas array were considered in the past, e. g. [4], [22], [23]. There are two essential constructions for Costas arrays, both of them form doubly periodic DDCs.…”

“…There are various papers, e.g. [3], [4], [17] in which an one-dimensional sequence (as a Sidon sequence or a ruler) is transformed into a two-dimensional synchronization pattern. The main goal of this paper is to establish the inverse transformation, in which a twodimensional synchronization pattern is transformed into a Sidon sequence, which is a one-dimensional sequence.…”

Design and quality metrics of point patterns for coded structured light illumination with diffractive optical elements in optical 3D sensors

“…10,[13][14][15][16] The construction of 2D toroidal PSMs not based on De Bruin sequences is more difficult. Several approaches have been suggested using evolutionary algorithms 14,17 or brute force exhaustive search.…”