Bounds for Single-Error-Correcting Analog Codes

2026-06-01

AI Summary

This study investigates the performance limits of real-valued linear analog codes with redundancy two under single-error correction, focusing on the minimum separation Γ₂(𝒞) required to distinguish large outliers from bounded perturbations. By integrating geometric characterizations via zonotopes with cyclic sine-product inequalities, the work resolves an open problem posed by Roth regarding the optimality of [n, n−2] codes and extends the result to any fixed redundancy r. The main contributions include proving that every [n, n−2] real linear code satisfies Γ₂(𝒞) ≥ 1/sin²(π/2n), and constructing a family of high-redundancy codes that achieve Γ₂(𝒞) = O(n^{1+1/(r−1)}). These results reveal the scaling law governing how separation grows with code length and redundancy.

Abstract

We study single-error correction for analog codes over $\mathbb{R}$. A key performance measure is the parameter $Γ_2(\mathcal{C})$, which quantifies the minimum separation required between large outlying errors that need to be located/corrected and bounded tolerable perturbations. We prove that every real linear $[n,n-2]$ code $\mathcal{C}$ satisfies \[ Γ_2(\mathcal{C})\ge \frac{1}{\sin^2(π/2n)}. \] This resolves Roth's open problem on the optimality of redundancy-two single-error-correcting analog codes. Our proof combines a zonotope-based geometric characterization of $Γ_2(\mathcal{C})$ with a cyclic sine-product inequality. We also construct analog codes with higher fixed redundancy and show that, for every fixed $r\ge 2$, there exists a class of real linear $[n,n-r]$ codes such that \[ Γ_2(\mathcal{C})\le O\left(n^{1+\frac{1}{r-1}}\right). \]

Problem

• analog codes
• single-error correction
• error separation
• minimum distance
• real linear codes

Innovation

• analog codes
• single-error correction
• zonotope
• redundancy
• error separation