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

Diameter Perfect Lee Codes

Lee codes have been intensively studied for more than 40 years. Interest in these codes has been triggered by the Golomb-Welch conjecture on the existence of the perfect error-correcting Lee codes. In this paper we deal with the existence and enumeration of diameter perfect Lee codes. As main results we determine all $q$ for which there exists a linear diameter-4 perfect Lee code of word length $n$ over $Z_{q},$ and prove that for each $n\geq 3$ there are uncountable many diameter-4 perfect Lee codes of word length $n$ over $Z.$ This is in a strict contrast with perfect error-correcting Lee codes of word length $n$ over $Z\,$\ as there is a unique such code for $n=3,$ and its is conjectured that this is always the case when $2n+1$ is a prime. We produce diameter perfect Lee codes by an algebraic construction that is based on a group homomorphism. This will allow us to design an efficient algorithm for their decoding. We hope that this construction will turn out to be useful far beyond the scope of this paper

Error-Correcting Codes and Minkowski’s Conjecture

The goal of this paper is twofold. The main one is to survey the latest results on the perfect and quasi-perfect Lee error correcting codes. The other goal is to show that the area of Lee error correcting codes, like many ideas in mathematics, can trace its roots to the Phytagorean theorem a2+b2 = c2. Thus to show that the area of the perfect Lee error correcting codes is an integral part of mathematics. It turns out that Minkowski’s conjecture, which is an interface of number theory, approximation theory, geometry, linear algebra, and group theory is one of the milestones on the route to Lee codes

Tilings by (0.5,n)(0.5,n)-Crosses and Perfect Codes

The existence question for tiling of the $n$-dimensional Euclidian space by crosses is well known. A few existence and nonexistence results are known in the literature. Of special interest are tilings of the Euclidian space by crosses with arms of length one, known also as Lee spheres with radius one. Such a tiling forms a perfect code. In this paper crosses with arms of length half are considered. These crosses are scaled by two to form a discrete shape. We prove that an integer tiling for such a shape exists if and only if $n=2^t-1$ or $n=3^t-1$, $t>0$. A strong connection of these tilings to binary and ternary perfect codes in the Hamming scheme is shown

On almost perfect linear Lee codes of packing radius 2

More than 50 years ago, Golomb and Welch conjectured that there is no perfect Lee codes $C$ of packing radius $r$ in $\mathbb{Z}^{n}$ for $r\geq2$ and $n\geq 3$. Recently, Leung and the second author proved that if $C$ is linear, then the Golomb-Welch conjecture is valid for $r=2$ and $n\geq 3$. In this paper, we consider the classification of linear Lee codes with the second-best possibility, that is the density of the lattice packing of $\mathbb{Z}^n$ by Lee spheres $S(n,r)$ equals $\frac{|S(n,r)|}{|S(n,r)|+1}$. We show that, for $r=2$ and $n\equiv 0,3,4 \pmod{6}$, this packing density can never be achieved.Comment: The extended abstract of an earlier version of this paper was presented in the 12th International Workshop on Coding and Cryptography (WCC) 202

Almost perfect linear Lee codes of packing radius 2 only exist for small dimensions

It is conjectured by Golomb and Welch around half a century ago that there is no perfect Lee codes $C$ of packing radius $r$ in $\mathbb{Z}^{n}$ for $r\geq2$ and $n\geq 3$. Recently, Leung and the second author proved this conjecture for linear Lee codes with $r=2$. A natural question is whether it is possible to classify the second best, i.e., almost perfect linear Lee codes of packing radius $2$. We show that if such codes exist in $\mathbb{Z}^n$, then $n$ must be $1,2, 11, 29, 47, 56, 67, 79, 104, 121, 134$ or $191$

Rainbow Perfect Domination in Lattice Graphs

Let 0<n\in\mathbb{Z}. In the unit distance graph of $\mathbb{Z}^n\subset\mathbb{R}^n$, a perfect dominating set is understood as having induced components not necessarily trivial. A modification of that is proposed: a rainbow perfect dominating set, or RPDS, imitates a perfect-distance dominating set via a truncated metric; this has a distance involving at most once each coordinate direction taken as an edge color. Then, lattice-like RPDS s are built with their induced components C having: {i} vertex sets V(C) whose convex hulls are n-parallelotopes (resp., both (n-1)- and 0-cubes) and {ii} each V(C) contained in a corresponding rainbow sphere centered at C with radius n (resp., radii 1 and n-2)