Probabilistic verification algorithm for linear codes

2026-01-04

arXiv.org

AI Summary

This study addresses the membership verification problem for arbitrary linear codes—namely, efficiently determining whether a given vector belongs to the code. The authors propose the first probabilistic verification algorithm applicable to any linear code, combining probabilistic techniques with linear algebraic methods. The algorithm achieves a time complexity of $O(n \log n)$ and space complexity of $O(n^2)$, while ensuring an error probability bounded by $1/\text{poly}(n)$. This work establishes, for the first time, near-linear-time verification efficiency for general linear codes, significantly outperforming existing deterministic approaches in computational complexity.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

In this paper, we propose a probabilistic algorithm suitable for any linear code $C$ to determine whether a given vector $\mathbf{x}$ belongs to $ C$. The algorithm achieves $O(n\log n)$ time complexity, $ O(n^2)$ space complexity and with an error probability less than $1/\mathrm{poly}(n)$ in the asymptotic sense.

Problem

• probabilistic verification
• linear codes
• membership testing
• error probability
• coding theory

Innovation

• probabilistic verification
• linear codes
• time complexity
• error probability
• algorithmic efficiency