  Several new applications and a number of new mathematical techniques have
increased the research on error-correcting codes in the Lee metric in the last
decade. In this work we consider several coding problems and constructions of
error-correcting codes in the Lee metric. First, we consider constructions of
dense error-correcting codes in relatively small dimensions over small
alphabets. The second problem we solve is construction of diametric perfect
codes with minimum distance four. We will construct such codes over various
lengths and alphabet sizes. The third problem is to transfer an n-dimensional
Lee sphere with large radius into a shape, with the same volume, located in a
relatively small box. Hadamard matrices play an essential role in the solutions
for all three problems. A construction of codes based on Hadamard matrices will
start our discussion. These codes approach the sphere packing bound for very
high rate range and appear to be the best known codes over some sets of
parameters.
