Unbounded Error Correcting Codes
Klim Efremenko, Or Zamir
TL;DR
This work introduces unbounded error-correcting codes to handle indefinite-length transmissions by defining $(R,\varepsilon)$-unbounded codes whose prefixes remain decodable under adversarial noise. It establishes a near-tight rate-distance tradeoff over binary alphabets, proving $R<1-\Omega(\sqrt{\varepsilon})$ and $R>1- O(\sqrt{\varepsilon \log\log(1/\varepsilon)})$; it also shows linear codes perform strictly worse than non-linear ones and that, in the random-noise regime, the optimal rate matches that of standard ECCs. The paper introduces subset codes as a key combinatorial tool, enabling improved constructions that push the rate closer to 1 via layered redundancy and checksums. It also delineates a distinction between adversarial and random-noise models and discusses alphabet-size effects, leaving open the precise gap and explicit constructions as future directions. Overall, unbounded codes reveal fundamental differences from classical ECCs in the unlimited-length setting and open avenues for streaming and real-time communication under hostile error patterns.
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 $R<1-Ω(\sqrt{\varepsilon})$ 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 $R=1-Θ(\sqrt{\varepsilon \log(1/\varepsilon)})$. (3) In the setting of random noise, unbounded codes achieve the same optimal rate as standard ECCs, $R=1-Θ(\varepsilon \log(1/\varepsilon))$. These results demonstrate fundamental differences between standard and unbounded codes.