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

TL;DR

This paper extends Elias bounds to symmetric limited-magnitude error channels in $\\mathbb{Z}^n$, deriving new bounds on error correction capabilities and code density for different error magnitudes.

Contribution

It introduces geometric bounds for perfect codes in the symmetric limited-magnitude error channel, distinguishing regimes based on error magnitude and providing bounds on error correction and code density.

Findings

For small error magnitudes ($s=1,2$), $e=O(\,\sqrt{n \,\log n})$.

For larger magnitudes ($s \\geq 3$), $e < \\sqrt{12.36 n}$.

Upper bound on packing density inversely proportional to error magnitude $s$.

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 $\Omega(\sqrt{n})$ number of errors, the density is bounded by a factor inversely proportional to the error magnitude $s$.