130 research outputs found
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
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
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
- …