On the Height Profile of Analog Error-Correcting Codes

2026-02-23

AI Summary

This work investigates the ability of linear codes in the analog domain to detect or localize large anomalous errors while tolerating small perturbations. To this end, it introduces the novel concept of a “height profile,” which precisely characterizes a code’s performance across error magnitudes and reveals its intrinsic connection to code structure. By leveraging optimization models, extremal combinatorial formulations, and geometric interpretations, the study establishes multiple equivalent characterizations of the height profile, thereby overcoming the limitations of classical minimum-distance analysis. Combining linear programming, combinatorial optimization, and structural properties of generator matrices, the paper develops explicit computational methods for several code families and quantitatively uncovers the relationship between the Δ/δ ratio and the number of correctable anomalies.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Cognitive Modeling & Cognitive Systems: Analogy
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

In recent work, it has been shown that maintaining reliability in analog vector--matrix multipliers can be modeled as the following coding problem. Vectors in $\mathbb{R}^k$ are encoded into codewords of a linear $[n,k,d]$ code $C$ over $\mathbb{R}$. For prescribed positive reals $\delta<\Delta$, additive errors of magnitude at most $\delta$ are tolerable and need no handling, yet outlying errors of magnitude greater than $\Delta$ are to be located or detected. The trade-off between the ratio $\Delta/\delta$ and the number of outlying errors that can be handled is determined by the height profile of $C$; as such, the height profile provides a finer description of the error handling capability of $C$, compared to the minimum distance $d$, which only determines the number of correctable errors. This work contains a further study of the notion of the height profile. Several characterizations of the height profile are presented, thereby yielding methods for computing it. The starting point is formulating this computation as an optimization problem that is solved by a set of linear programs. This, in turn, leads to a combinatorial characterization of the height profile as a maximum (or max--min) over a certain finite set of codewords of $C$. Moreover, this characterization is shown to have a simple geometric interpretation when the columns of the generator matrix of $C$ all have the same $L_2$ norm. Through examples of several code families, it is demonstrated how the results herein can be used to compute the height profile explicitly.

Problem

• analog error-correcting codes
• height profile
• outlying errors
• error detection
• reliability

Innovation

• height profile
• analog error-correcting codes
• linear programming
• combinatorial characterization
• error tolerance