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

Abstract

Limited-magnitude errors modify a transmitted integer vector in at most $t$ entries, where each entry can increase by at most $\kp$ or decrease by at most $\km$. 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 $\cB(n,t,\kp,\km)$. In this paper, we focus on the case where $t=2$ and $\km=\kp-1$, and we derive necessary conditions on $m$ and $n$ for the existence of a lattice tiling of $\cB(n,2,m,m-1)$. 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 $\cB(n,2,2,1)$ or $\cB(n,2,3,2)$ exists for any $n\ge 3$.