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

Quasi-Perfect Codes From Cayley Graphs Over Integer Rings

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)The problem of searching for perfect codes has attracted great attention since the paper by Golomb and Welch, in which the existence of these codes over Lee metric spaces was considered. Since perfect codes are not very common, the problem of searching for quasi-perfect codes is also of great interest. In this aspect, also quasi-perfect Lee codes have been considered for 2-D and 3-D Lee metric spaces. In this paper, constructive methods for obtaining quasi-perfect codes over metric spaces modeled by means of Gaussian and Eisenstein-Jacobi integers are given. The obtained codes form ideals of the integer ring thus preserving the property of being geometrically uniform codes. Moreover, they are able to correct more error patterns than the perfect codes which may properly be used in asymmetric channels. Therefore, the results in this paper complement the constructions of perfect codes previously done for the same integer rings. Finally, decoding algorithms for the quasi-perfect codes obtained in this paper are provided and the relationship of the codes and the Lee metric ones is investigated.59959055916Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Spanish Ministry of Science [TIN2010-21291-C02-02, AP2010-4900, CSD2007-00050]European HiPEAC Network of ExcellenceFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)FAPESP [2007/56052-8]CNPq [303059/2010-9]Spanish Ministry of Science [TIN2010-21291-C02-02, AP2010-4900, CSD2007-00050