12 research outputs found
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
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 linear code was recently introduced. Codes attaining the
established upper bound are the Maximum Weight Spectrum (MWS) codes. Those
codes with the same weight set as are called Full
Weight Spectrum (FWS) codes. FWS codes are necessarily ``short", whereas MWS
codes are necessarily ``long". For fixed the values of for which
an -FWS code exists are completely determined, but the determination
of the minimum length of an -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 for which an FWS code
exists, and bounds on 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 (the minimum length of Lee MWS codes),
and pose the determination of 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
A New Approach Towards the Golomb-Welch Conjecture
The Golomb-Welch conjecture deals with the existence of perfect % -error
correcting Lee codes of word length 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 . Using this new approach
we also construct the first quasi-perfect Lee codes for dimension and
show that, for fixed , there are only finitely many such codes over
Rainbow Perfect Domination in Lattice Graphs
Let 0<n\in\mathbb{Z}. In the unit distance graph of , 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)
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
Quasi-Perfect Lee Codes of Radius 2 and Arbitrarily Large Dimension
A construction of two-quasi-perfect Lee codes is given over the space ?np for p prime, p ? ±5 (mod 12), and n = 2[p/4]. It is known that there are infinitely many such primes. Golomb and Welch conjectured that perfect codes for the Lee metric do not exist for dimension n ? 3 and radius r ? 2. This conjecture was proved to be true for large radii as well as for low dimensions. The codes found are very close to be perfect, which exhibits the hardness of the conjecture. A series of computations show that related graphs are Ramanujan, which could provide further connections between coding and graph theories
On Grid Codes
If is finite alphabet for , the Manhattan distance is
defined in . A grid code is introduced as a subset of
. Alternative versions of the Hamming and
Gilbert-Varshamov bounds are presented for grid codes. If is a cyclic
group for , some bounds for the minimum Manhattan distance of codes
that are cyclic subgroups of are determined in terms of
their minimum Hamming and Lee distances. Examples illustrating the main results
are provided