On lattice tilings of ${\mathbb{Z}}^n$ by limited magnitude error balls $\mathcal{B}(n,2,k_1,k_2)$ with $k_1>k_2$

Lattice tilings of ${\mathbb{Z}}^n$ by limited-magnitude error balls correspond to linear perfect codes under such error models and play a crucial role in flash memory applications. In this work, we establish three main results. First, we fully determine the existence of lattice tilings by $\mathcal{B}(n,2,3,0)$ in all dimensions $n$. Second, we completely resolve the case $k_1=k_2+1$. Finally, we prove that for any integers $k_1>k_2\ge 0$ where $k_1+k_2+1$ is composite, no lattice tiling of ${\mathbb{Z}}^n$ by the error ball $\mathcal{B}(n,2,k_1,k_2)$ exists for sufficiently large $n$.