  A well known conjecture of Golomb and Welch is that the only nontrivial
perfect codes in the Lee and Manhattan metrics have length two or minimum
distance three. This problem and related topics were subject for extensive
research in the last forty years. In this paper two product constructions for
perfect Lee codes and diameter perfect Lee codes are presented. These
constructions yield a large number of nonlinear perfect codes and nonlinear
diameter perfect codes in the Lee and Manhattan metrics. A short survey and
other related problems on perfect codes in the Lee and the Manhattan metrics
are also discussed.
