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

2026-01-20

AI Summary

This study investigates the existence of perfect error-correcting codes in symmetric bounded-magnitude error channels, where up to $e$ coordinates suffer integer errors of magnitude at most $s$. By generalizing the Elias bound and integrating geometric tiling theory with combinatorial analysis, the work characterizes the tiling properties of error spheres in integer lattice spaces without assuming any underlying lattice structure. For the first time, asymptotic bounds on $e$ are established for varying $s$: specifically, $e = O(\sqrt{n \log n})$ when $s = 1,2$, and $e < \sqrt{12.36n}$ when $s \geq 3$. The results yield new necessary conditions for the existence of perfect codes, an upper bound on the packing density of non-perfect codes, and reveal an inverse relationship between error-correction capability and error magnitude $s$.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Information Theory
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

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 $s$.

Problem

• perfect codes
• symmetric limited-magnitude error channel
• tilings
• Elias bound
• error-correcting codes

Innovation

• Elias bound
• symmetric limited-magnitude error
• perfect codes
• tiling of Z^n
• packing density