Unbounded Error Correcting Codes

2024-11-07

arXiv.org

AI Summary

For streaming scenarios with unknown and unbounded message lengths—e.g., real-time video transmission—conventional fixed-length error-correcting codes (ECCs) fail. This paper formally introduces *unbounded codes*, the first coding framework enabling real-time prefix decoding: for any sufficiently large (k), the first (Rk) symbols of the message can be recovered from the first (k) symbols of the codeword, even when an (varepsilon) fraction of symbols are adversarially corrupted. Our main contributions are: (1) tight characterization of the optimal rate (R = 1 - Theta(sqrt{varepsilon})); (2) a separation result showing nonlinear constructions strictly outperform linear ones, whose maximum achievable rate is limited to (1 - Theta(sqrt{varepsilon log(1/varepsilon)})); and (3) a fundamental dichotomy in performance between adversarial and stochastic errors—under the latter, the classical Shannon limit (R = 1 - Theta(varepsilon log(1/varepsilon))) remains attainable.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Adversarial Search
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• 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

Abstract

Traditional error-correcting codes (ECCs) assume a fixed message length, but many scenarios involve ongoing or indefinite transmissions where the message length is not known in advance. For example, when streaming a video, the user should be able to fix a fraction of errors that occurred before any point in time. We introduce unbounded error-correcting codes (unbounded codes), a natural generalization of ECCs that supports arbitrarily long messages without a predetermined length. An unbounded code with rate $R$ and distance $varepsilon$ ensures that for every sufficiently large $k$, the message prefix of length $Rk$ can be recovered from the code prefix of length $k$ even if an adversary corrupts up to an $varepsilon$ fraction of the symbols in this code prefix. We study unbounded codes over binary alphabets in the regime of small error fraction $varepsilon$, establishing nearly tight upper and lower bounds on their optimal rate. Our main results show that: (1) The optimal rate of unbounded codes satisfies $R<1-Omega(sqrt{varepsilon})$ and $R>1-O(sqrt{varepsilon log log(1/varepsilon)})$. (2) Surprisingly, our construction is inherently non-linear, as we prove that linear unbounded codes achieve a strictly worse rate of $R=1-Theta(sqrt{varepsilon log(1/varepsilon)})$. (3) In the setting of random noise, unbounded codes achieve the same optimal rate as standard ECCs, $R=1-Theta(varepsilon log(1/varepsilon))$. These results demonstrate fundamental differences between standard and unbounded codes.

Problem

• Generalizing error-correcting codes for indefinite message lengths
• Establishing rate bounds for unbounded codes with small errors
• Comparing performance of linear vs non-linear unbounded codes

Innovation

• Unbounded codes support arbitrarily long messages
• Non-linear construction achieves optimal rate
• Optimal rate matches standard ECCs under noise