Interactive Error Correcting Codes Over Binary Erasure Channels Resilient to $>\frac12$ Adversarial Corruption

TL;DR

This paper introduces interactive error correcting codes that surpass the traditional 50% error resilience limit for adversarial erasures, achieving up to 60% resilience, and establishes upper bounds on possible resilience levels.

Contribution

The work constructs the first interactive error correcting codes resilient to more than half of adversarial erasures and proves fundamental upper bounds on their maximum possible resilience.

Findings

Achieves 60% adversarial erasure resilience with interactive codes.

Proves an upper bound of 66.7% for binary interactive erasure codes.

Establishes a 27% resilience limit for binary interactive bit-flip codes.

Abstract

An error correcting code ($\mathsf{ECC}$) allows a sender to send a message to a receiver such that even if a constant fraction of the communicated bits are corrupted, the receiver can still learn the message correctly. Due to their importance and fundamental nature, $\mathsf{ECC}$s have been extensively studied, one of the main goals being to maximize the fraction of errors that the $\mathsf{ECC}$ is resilient to. For adversarial erasure errors (over a binary channel) the maximal error resilience of an $\mathsf{ECC}$ is $\frac12$ of the communicated bits. In this work, we break this $\frac12$ barrier by introducing the notion of an interactive error correcting code ($\mathsf{iECC}$) and constructing an $\mathsf{iECC}$ that is resilient to adversarial erasure of $\frac35$ of the total communicated bits. We emphasize that the adversary can corrupt both the sending party and the receiving party, and that both parties' rounds contribute to the adversary's budget. We also prove an impossibility (upper) bound of $\frac23$ on the maximal resilience of any binary $\mathsf{iECC}$ to adversarial erasures. In the bit flip setting, we prove an impossibility bound of $\frac27$.