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 work investigates lattice tilings of $\mathbb{Z}^n$ by limited-magnitude balls $\mathcal{B}(n,2,k_{1},k_{2})$, which correspond to linear perfect codes under constrained error models. Using a group-ring framework, it translates tiling existence into the existence of a finite abelian group $G$ and a homomorphism $\phi: \mathbb{Z}^n \to G$ that bijectively maps the ball to $G$, enabling precise combinatorial counters $\psi(m,i)$ and $\psi(m,i,j)$ to constrain intersections. The paper delivers three main results: a complete classification for $\mathcal{B}(n,2,3,0)$ (existence only for $n=3$ with an explicit tiling in $\mathbb{Z}_{37}$), a complete resolution for the case $k_{1}=k_{2}+1$ (no lattice tilings for any $n\ge 3$ and $k\ge 2$), and a general nonexistence theorem for composite $k_{1}+k_{2}+1$ that provides explicit bounds on $n$ beyond which tilings cannot occur. These results advance the understanding of exact tilings in higher dimensions with two-coordinate errors and have implications for designing optimal, error-resilient data representations in flash-memory systems.

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 $n$.

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 2.1
• Definition 2.2
• Theorem 2.3
• proof
• Definition 2.4
• Lemma 2.5