Tight Lower Bounds on The Single-Error Detection Threshold for Analog Error-Correcting Codes

2026-05-09

AI Summary

This study investigates lower bounds on the threshold for linear analog error-correcting codes over the real numbers in single-error detection scenarios, addressing two open problems posed by Roth concerning subspace height profiles. By integrating techniques from convex optimization, linear algebra, and height profile analysis, the work provides the first affirmative resolution to Problem 2 and extends this result to the case where \(n - k\) divides \(k\), thereby establishing the tightness of the lower bound in Problem 1. Notably, when \(n\) is even, it is shown that every \((n-2)\)-dimensional real subspace necessarily contains a nonzero vector satisfying a specific amplitude-ratio condition, thus filling a critical gap in the theoretical understanding of single-error detection thresholds.

Abstract

Analog error-correcting codes (Analog ECCs) for approximate vector-matrix multiplication have been extensively studied as means to achieve fault-tolerant in-memory computation. The theoretical foundations for such coding schemes, particularly the characterization of their correction capabilities via the height profile, have been well established in recent literature. In this paper, we focus on the case of single-error detection Analog ECCs. Among several open problems related to this case proposed by Ron M. Roth in [1], Problem 1 asks: "Identify the values of $k$ and $n$ for which every linear $[n, k]$ code $\mathcal{C}$ over $\mathbb{R}$ satisfies: $$\mathsf{h}_1(\mathcal{C}):=\max_{\boldsymbol{c}\in \mathcal{C}\setminus{\{\boldsymbol{0}\}}}\mathsf{h}_1(\boldsymbol{c})\geq \Big\lceil \frac{k}{n-k} \Big\rceil.\text{"}$$ Here, for any $\boldsymbol{x}\in\mathbb{R}^n$, $\mathsf{h}_1(\boldsymbol{x})$ represents the ratio between the largest and second largest absolute values of $\boldsymbol{x}$'s entries. As the simplest special case of Problem 1 (with $n-k=2$), the following problem was posed as Problem 2 in [1]: "Must every $(n-2)$-dimensional subspace of $\mathbb{R}^n$, $n$ even, contain a nonzero vector in which the ratio between the largest and second largest absolute values of its entries is at least $(n/2)-1$?" These problems directly pertain to the lower bounds on the single-error detection threshold for Analog ECCs: Problem 1 corresponds to arbitrary $n-k$ and Problem 2 corresponds to $n-k=2$. In this paper, we provide an affirmative answer to Problem 2 and a rigorous proof using theories related to convex optimization. Furthermore, we extend our analytical method to show that the lower bound in Problem 1 is tight for the case where $n-k$ divides $k$. Our results fill the gap in the lower bound theory of thresholds for single-error detection in Analog ECCs.

Problem

• Analog ECCs
• single-error detection
• threshold lower bound
• height profile
• linear codes

Innovation

• Analog error-correcting codes
• single-error detection
• height profile
• convex optimization
• tight lower bounds