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.

翻译：本文研究长度为 $n$ 在剩余环 $\mathbb{Z}_m := \mathbb{Z}/m\mathbb{Z}$ 上的 $t$-Lee-纠错码，并确定其上界和下界。我们采用两种不同的方法：估计二部图上的孤立节点以及图容器方法。前者给出固定大小码的密度结果，后者则适用于任意大小码。这证实了近期关于线性 Lee 度量码的密度结果，并为非线性码提供了新的密度结论。为应用图容器算法的一个变体，我们还研究了 Lee 度量中球的一些几何性质。