Quasi-Cross Lattice Tilings with Applications to Flash Memory

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 $\mathcal{B}(n,2,1,1)$.

On the Non-existence of Lattice Tilings by Quasi-crosses

We study necessary conditions for the existence of lattice tilings of $\R^n$ by quasi-crosses. We prove non-existence results, and focus in particular on the two smallest unclassified shapes, the $(3,1,n)$-quasi-cross and the $(3,2,n)$-quasi-cross. We show that for dimensions $n\leq 250$, apart from the known constructions, there are no lattice tilings of $\R^n$ by $(3,1,n)$-quasi-crosses except for ten remaining cases, and no lattice tilings of $\R^n$ by $(3,2,n)$-quasi-crosses except for eleven remaining cases.

关于当 $k_1>k_2$ 时由有限幅度错误球 $\mathcal{B}(n,2,k_{1},k_{2})$ 对 $\mathbb{Z}^n$ 进行格铺砌的研究

On lattice tilings of $\mathbb{Z}^n$ by limited magnitude error balls $\mathcal{B}(n,2,k_{1},k_{2})$ with $k_1>k_2$

本文研究有限幅度错误球对整数格的格铺砌，完全刻画了 $\mathcal{B}(n,2,3,0)$ 与 $k_1=k_2+1$ 情形，并证明合数条件下高维不存在铺砌。

On Tilings of Asymmetric Limited-Magnitude Balls

We study whether an asymmetric limited-magnitude ball may tile $\mathbb{Z}^n$. This ball generalizes previously studied shapes: crosses, semi-crosses, and quasi-crosses. Such tilings act as perfect error-correcting codes in a channel which changes a transmitted integer vector in a bounded number of entries by limited-magnitude errors. A construction of lattice tilings based on perfect codes in the Hamming metric is given. Several non-existence results are proved, both for general tilings, and lattice tilings. A complete classification of lattice tilings for two certain cases is proved.