Interactive Coding with Unbounded Noise

TL;DR

The paper tackles interactive coding under unbounded and adversarial noise by introducing the Unbounded Probabilistic Erasure-Flip (UPEF) model and a modified Oblivious Flag framework. It develops two complementary constructions: (i) a UPEF-optimal coding scheme via code concatenation that achieves $O(N+T)$ communication and failure probability $2^{- ext{Ω}(N)}$, and (ii) a UF-optimal scheme obtained by translating the UPEF construction into the standard unbounded-flip model, aided by constant-size AMD codes. A key methodological innovation is the iterative approach with fixed flip probabilities (e.g., $p_i=2/3$) and a progressive, self-tuning execution that terminates once the noise level is sufficiently low; the approach leverages inner erasure-flip protection and outer synchronization tools (e.g., HSV18) to preserve correctness. The results significantly extend interactive coding to settings with arbitrary and unknown noise levels while preserving near-optimal communication and exponentially small failure, providing practical means for robust distributed computation under extreme channel interference. Overall, the work tightens the bridge between theoretical models of oblivious, unbounded noise and practical, near-optimal coding schemes that withstand severe channel corruption.

Abstract

Interactive coding allows two parties to conduct a distributed computation despite noise corrupting a certain fraction of their communication. Dani et al.\@ (Inf.\@ and Comp., 2018) suggested a novel setting in which the amount of noise is unbounded and can significantly exceed the length of the (noise-free) computation. While no solution is possible in the worst case, under the restriction of oblivious noise, Dani et al.\@ designed a coding scheme that succeeds with a polynomially small failure probability. We revisit the question of conducting computations under this harsh type of noise and devise a computationally-efficient coding scheme that guarantees the success of the computation, except with an exponentially small probability. This higher degree of correctness matches the case of coding schemes with a bounded fraction of noise. Our simulation of an $N$-bit noise-free computation in the presence of $T$ corruptions, communicates an optimal number of $O(N+T)$ bits and succeeds with probability $1-2^{-Ω(N)}$. We design this coding scheme by introducing an intermediary noise model, where an oblivious adversary can choose the locations of corruptions in a worst-case manner, but the effect of each corruption is random: the noise either flips the transmission with some probability or otherwise erases it. This randomized abstraction turns out to be instrumental in achieving an optimal coding scheme.

Theorems & Definitions (75)

• Theorem 1.1: Main
• Theorem 2.1: Theorem 1.2 in HSV18
• Theorem 3.1
• Theorem 4.1
• Lemma 4.2
• proof
• proof
• Lemma 4.4
• proof
• Lemma 4.5