On lattice tilings of $\mathbb{Z}^n$ by limited magnitude error balls $\mathcal{B}(n,2,k_{1},k_{2})$ with $k_1>k_2$

TL;DR

This paper investigates the conditions under which lattice tilings of integer lattices by limited-magnitude error balls exist, providing complete classifications for specific cases and showing non-existence in others, which advances understanding of perfect codes in error correction.

Contribution

The paper fully characterizes lattice tilings by certain error balls in all dimensions and resolves specific cases, contributing new theoretical results to coding theory and lattice tiling.

Findings

Complete classification for $ ext{B}(n,2,3,0)$ tilings.

Resolution of the case $k_1=k_2+1$.

Non-existence results for composite $k_1+k_2+1$ in large dimensions.

Abstract

Lattice tilings of $\mathbb{Z}^n$ by limited-magnitude error balls correspond to linear perfect codes under such error models and play a crucial role in flash memory applications. In this work, we establish three main results. First, we fully determine the existence of lattice tilings by $\mathcal{B}(n,2,3,0)$ in all dimensions $n$. Second, we completely resolve the case $k_1=k_2+1$. Finally, we prove that for any integers $k_1>k_2\ge0$ where $k_1+k_2+1$ is composite, no lattice tiling of $\mathbb{Z}^n$ by the error ball $\mathcal{B}(n,2,k_1,k_2)$ exists for sufficiently large $n$.