68 research outputs found
On the nonexistence of three-dimensional tiling in the Lee metric II
AbstractWe prove that there does not exist a tiling of R3 with Lee spheres of radius greater than 0 such that the radius of at least one of them is greater than one
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 Minkowskiâs 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 Minkowskiâs 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
Interleaving schemes for multidimensional cluster errors
We present two-dimensional and three-dimensional interleaving techniques for correcting two- and three-dimensional bursts (or clusters) of errors, where a cluster of errors is characterized by its area or volume. Correction of multidimensional error clusters is required in holographic storage, an emerging application of considerable importance. Our main contribution is the construction of efficient two-dimensional and three-dimensional interleaving schemes. The proposed schemes are based on t-interleaved arrays of integers, defined by the property that every connected component of area or volume t consists of distinct integers. In the two-dimensional case, our constructions are optimal: they have the lowest possible interleaving degree. That is, the resulting t-interleaved arrays contain the smallest possible number of distinct integers, hence minimizing the number of codewords required in an interleaving scheme. In general, we observe that the interleaving problem can be interpreted as a graph-coloring problem, and introduce the useful special class of lattice interleavers. We employ a result of Minkowski, dating back to 1904, to establish both upper and lower bounds on the interleaving degree of lattice interleavers in three dimensions. For the case tâĄ0 mod 6, the upper and lower bounds coincide, and the Minkowski lattice directly yields an optimal lattice interleaver. For tâ 0 mod 6, we construct efficient lattice interleavers using approximations of the Minkowski lattice
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 for which there exists a linear diameter-4 perfect Lee code
of word length over and prove that for each there are
uncountable many diameter-4 perfect Lee codes of word length over This
is in a strict contrast with perfect error-correcting Lee codes of word length
over \ as there is a unique such code for and its is
conjectured that this is always the case when 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
Tilings by -Crosses and Perfect Codes
The existence question for tiling of the -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
or , . A strong connection of these tilings to binary
and ternary perfect codes in the Hamming scheme is shown
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