On the Number of $t$-Lee-Error-Correcting Codes

We consider $t$-Lee-error-correcting codes of length $n$ over the residue ring $\mathbb{Z}_m := \mathbb{Z}/m\mathbb{Z}$ and determine upper and lower bounds on the number of $t$-Lee-error-correcting codes. We use two different methods, namely estimating isolated nodes on bipartite graphs and the graph container method. The former gives density results for codes of fixed size and the latter for any size. This confirms some recent density results for linear Lee metric codes and provides new density results for nonlinear codes. To apply a variant of the graph container algorithm we also investigate some geometrical properties of the balls in the Lee metric.