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

TL;DR

This paper investigates the conditions under which lattice tilings of asymmetric limited-magnitude error balls exist, providing bounds on parameters for such tilings in the context of error-correcting codes.

Contribution

It derives necessary conditions and bounds for the existence of lattice tilings of specific asymmetric limited-magnitude balls, advancing understanding of perfect codes for these error models.

Findings

Necessary bounds on parameters for tilings when t=2 and km=kp-1.

No lattice tilings exist for certain small cases with m=2 or 3.

Provides inequalities relating n and m for the existence of tilings.

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 $n\geq 3$.