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

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

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

A New Approach Towards the Golomb-Welch Conjecture

The Golomb-Welch conjecture deals with the existence of perfect $e$% -error correcting Lee codes of word length $n,$ $PL(n,e)$ codes. Although there are many papers on the topic, the conjecture is still far from being solved. In this paper we initiate the study of an invariant connected to abelian groups that enables us to reformulate the conjecture, and then to prove the non-existence of linear PL(n,2) codes for $n\leq 12$. Using this new approach we also construct the first quasi-perfect Lee codes for dimension $n=3,$ and show that, for fixed $n$, there are only finitely many such codes over $Z^n$

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

Block Codes on Pomset Metric

Given a regular multiset $M$ on $[n]=\{1,2,\ldots,n\}$, a partial order $R$ on $M$, and a label map $\pi : [n] \rightarrow \mathbb{N}$ defined by $\pi(i) = k_i$ with $\sum_{i=1}^{n}\pi (i) = N$, we define a pomset block metric $d_{(Pm,\pi)}$ on the direct sum $\mathbb{Z}_{m}^{k_1} \oplus \mathbb{Z}_{m}^{k_2} \oplus \ldots \oplus \mathbb{Z}_{m}^{k_n}$ of $\mathbb{Z}_{m}^{N}$ based on the pomset $\mathbb{P}=(M,R)$. This pomset block metric $d_{(Pm,\pi)}$ extends the classical pomset metric which accommodate Lee metric introduced by I. G. Sudha and R. S. Selvaraj, in particular, and generalizes the poset block metric introduced by M. M. S. Alves et al, in general, over $\mathbb{Z}_m$. We find $I$-perfect pomset block codes for both ideals with partial and full counts. Further, we determine the complete weight distribution for $(P,\pi)$-space, thereby obtaining it for $(P,w)$-space, and pomset space, over $\mathbb{Z}_m$. For chain pomset, packing radius and Singleton type bound are established for block codes, and the relation of MDS codes with $I$-perfect codes is investigated. Moreover, we also determine the duality theorem of an MDS $(P,\pi)$-code when all the blocks have the same length.Comment: 17 Page

Optimal Interleaving on Tori

We study t-interleaving on two-dimensional tori, which is defined by the property that any connected subgraph with t or fewer vertices in the torus is labelled by all distinct integers. It has applications in distributed data storage and burst error correction, and is closely related to Lee metric codes. We say that a torus can be perfectly t-interleaved if its t-interleaving number – the minimum number of distinct integers needed to t-interleave the torus – meets the spherepacking lower bound. We prove the necessary and sufficient conditions for tori that can be perfectly t-interleaved, and present efficient perfect t-interleaving constructions. The most important contribution of this paper is to prove that the t-interleaving numbers of tori large enough in both dimensions, which constitute by far the majority of all existing cases, is at most one more than the sphere-packing lower bound, and to present an optimal and efficient t-interleaving scheme for them. Then we prove some bounds on the t-interleaving numbers for other cases, completing a general picture for the t-interleaving problem on 2-dimensional tori