Perfect Codes over Lipschitz Integers

Cayley graphs over quotients of the quaternion integers are going to be used to define a new metric over four dimensional lattices. We will consider perfect 1-error correcting codes according to this metric space. We will show that, in some cases, these lattices can be represented as two-dimensional constellations, which allow us to state a relation between the Lee metric and this new Lipschitz metric.