Unbounded Error Correcting Codes

TL;DR

This paper introduces unbounded error-correcting codes that support indefinite message lengths, providing nearly optimal rates and revealing fundamental differences from traditional ECCs, especially in non-linear constructions.

Contribution

The paper defines unbounded codes for ongoing transmissions, establishes tight bounds on their rates, and shows their non-linear nature is essential for optimal performance.

Findings

Optimal rate bounds: R<1−Ω(√ε) and R>1−O(√(ε log log(1/ε)))

Linear unbounded codes have worse rate R=1−Θ(√(ε log(1/ε)))

In random noise, unbounded codes match standard ECC rates R=1−Θ(ε log(1/ε))

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.