Elias-type Bounds for Codes in the Symmetric Limited-Magnitude Error Channel

TL;DR

This paper analyzes the existence of perfect codes in ${\mathbb Z}^n$ under symmetric limited-magnitude errors by recasting the problem as tilings with the error ball ${V(n,e,s)}$ and developing an Elias-type bound for the distance ${d_s}$. It establishes two regimes—${s\in\{1,2\}}$ yields ${e=O(\sqrt{n\log n})}$ (and stronger piecewise bounds), while ${s\ge 3}$ gives a sharp bound ${e<\sqrt{12.36 n}}$—without assuming lattice structure. The authors extend the approach to non-perfect codes, deriving packing-density upper bounds that decay with the magnitude ${s}$ when the error-correcting capability scales as ${e=\Omega(\sqrt{n})}$. These results generalize beyond lattice tilings, offering broad, quantitative constraints on the existence of perfect codes and guiding future constructions and nonexistence proofs in symmetric limited-magnitude channels.

Abstract

We study perfect error-correcting codes in $\mathbb{Z}^n$ for the symmetric limited-magnitude error channel, where at most $e$ coordinates of an integer vector may be altered by a value whose magnitude is at most $s$. Geometrically, such codes correspond to tilings of $\mathbb{Z}^n$ by the symmetric limited-magnitude error ball $\mathcal{B}(n,e,s,s)$. Given $n$ and $s$, we adapt the geometric ideas underlying the Elias bound for the Hamming metric to the distance $d_s$ tailed for this channel, and derive new necessary conditions on $e$ for the existence of perfect codes / tilings, without assuming any lattice structure. Our main results identify two distinct regimes depending on the error magnitude. For small error magnitudes ($s \in \{1, 2\}$), we prove that if the number of correctable errors does not exceed a certain fraction of $n$, then it is asymptotically bounded by $e = \mathcal{O}(\sqrt{n \log n})$. In contrast, for larger magnitudes ($s \geq 3$), we establish a significantly sharper bound of $e < \sqrt{12.36n}$, which holds without any restriction on $e$ being below a given fraction of $n$. Finally, by extending our method to non-perfect codes, we derive an upper bound on packing density, showing that for codes correcting a linear or $Ω(\sqrt{n})$ number of errors, the density is bounded by a factor inversely proportional to the error magnitude $s$.

Theorems & Definitions (23)

• Theorem 1.1: WeiSch22EJC
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Proposition 2.2
• Lemma 3.1
• Lemma 3.2
• Corollary 3.1
• Lemma 3.3