On Lattice Tilings of Asymmetric Limited-Magnitude Balls $\text{cB}(n,2,m,m-1)$

Limited-magnitude errors modify a transmitted integer vector in at most $t$ entries, where each entry can increase by at most $\text{kp}$ or decrease by at most $\text{km}$. This channel model is particularly relevant to applications such as flash memories and DNA storage. A perfect code for this channel is equivalent to a tiling of ${\mathbb{Z}}^n$ by asymmetric limited-magnitude balls $\text{cB}(n,t,\text{kp},\text{km})$. In this paper, we focus on the case where $t=2$ and $\text{km}=\text{kp}-1$, and we derive necessary conditions on $m$ and $n$ for the existence of a lattice tiling of $\text{cB}(n,2,m,m-1)$. Specifically, we prove that if such a tiling exists, then either $4\le m\le 512$ and $n<7.23m+4$, or $m>512$ and $n<4m$. In particular, for $m=2$ and $m=3$, we show that no lattice tiling of $\text{cB}(n,2,2,1)$ or $\text{cB}(n,2,3,2)$ exists for any $n\ge 3$.