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On Lattice Tilings of Asymmetric Limited-Magnitude Balls $\cB(n,2,m,m-1)$
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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 $\Z^n$ 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 $4\leq m \leq 512$ 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 $n\geq 3$.
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Cited by 1 Pith paper
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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 Z^n by error balls B(n,2,3,0) exist only for n=3, none exist for B(n,2,k,k-1), and for k1>k2 with k1+k2+1 composite no tiling exists in sufficiently high dimension.
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