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

TL;DR

This work addresses the existence of lattice tilings of the asymmetric limited-magnitude ball $\mathcal{B}(n,2,m,m-1)$, reframing tilings as $2$-splittings of finite Abelian groups. The authors derive two complementary bounds on $m_2(G)$ via counting techniques and projections, and combine these with a case analysis and computer search to obtain a concrete necessary condition: if a lattice tiling exists then either $4\le m\le 512$ with $n<7.23m+4$, or $m>512$ with $n<4m$. They also prove nonexistence for $m=2$ and $m=3$ with $n\ge 3$, using congruence arguments and the Nagell equation. The results significantly narrow the feasible parameter region for lattice tilings in this setting, informing both theory and potential applications in flash memory and DNA storage where limited-magnitude errors arise.

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 $n\geq 3$.

Theorems & Definitions (34)

• Theorem 1
• Definition 2
• Theorem 3: Lemma 4 and Corollary 1 in buzaglo2012tilings
• Theorem 4: Lemma 3 and Corollary 1 in buzaglo2012tilings
• Lemma 5
• proof
• Lemma 6: Nagell, nagell1961diophantine
• Lemma 7
• Lemma 8
• Lemma 9