Folding, Tiling, and Multidimensional Coding

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

50 Years of the Golomb--Welch Conjecture

Since 1968, when the Golomb--Welch conjecture was raised, it has become the main motive power behind the progress in the area of the perfect Lee codes. Although there is a vast literature on the topic and it is widely believed to be true, this conjecture is far from being solved. In this paper, we provide a survey of papers on the Golomb--Welch conjecture. Further, new results on Golomb--Welch conjecture dealing with perfect Lee codes of large radii are presented. Algebraic ways of tackling the conjecture in the future are discussed as well. Finally, a brief survey of research inspired by the conjecture is given.Comment: 28 pages, 2 figure

MWS and FWS Codes for Coordinate-Wise Weight Functions

A combinatorial problem concerning the maximum size of the (hamming) weight set of an $[n,k]_q$ linear code was recently introduced. Codes attaining the established upper bound are the Maximum Weight Spectrum (MWS) codes. Those $[n,k]_q$ codes with the same weight set as $\mathbb{F}_q^n$ are called Full Weight Spectrum (FWS) codes. FWS codes are necessarily ``short", whereas MWS codes are necessarily ``long". For fixed $k,q$ the values of $n$ for which an $[n,k]_q$-FWS code exists are completely determined, but the determination of the minimum length $M(H,k,q)$ of an $[n,k]_q$-MWS code remains an open problem. The current work broadens discussion first to general coordinate-wise weight functions, and then specifically to the Lee weight and a Manhattan like weight. In the general case we provide bounds on $n$ for which an FWS code exists, and bounds on $n$ for which an MWS code exists. When specializing to the Lee or to the Manhattan setting we are able to completely determine the parameters of FWS codes. As with the Hamming case, we are able to provide an upper bound on $M(\mathcal{L},k,q)$ (the minimum length of Lee MWS codes), and pose the determination of $M(\mathcal{L},k,q)$ as an open problem. On the other hand, with respect to the Manhattan weight we completely determine the parameters of MWS codes.Comment: 17 page