On lattice tilings of $\mathbb{Z}^{n}$ by limited magnitude error balls $\mathcal{B}(n,2,1,1)$

Limited magnitude error model has applications in flash memory. In this model, a perfect code is equivalent to a tiling of $\mathbb{Z}^n$ by limited magnitude error balls. In this paper, we give a complete classification of lattice tilings of $\mathbb{Z}^n$ by limited magnitude error balls $\mathcal{B}(n,2,1,1)$.