Efficient Interactive Coding Achieving Optimal Error Resilience Over the Binary Channel

TL;DR

This paper presents a new efficient interactive coding scheme over binary channels that achieves optimal error resilience of one-sixth of the communication, improving upon previous suboptimal or exponential complexity solutions.

Contribution

The authors introduce a novel layered code that enables positive rate, computational efficiency, and optimal error resilience in binary interactive communication.

Findings

Achieves optimal  -  adversarial error resilience

Constructs a positive rate, efficient protocol over binary channels

Introduces layered codes, a new coding structure for online encoding

Abstract

Given a noiseless protocol $\pi_0$ computing a function $f(x, y)$ of Alice and Bob's private inputs $x, y$, the goal of interactive coding is to construct an error-resilient protocol $\pi$ computing $f$ such that even if some fraction of the communication is adversarially corrupted, both parties still learn $f(x, y)$. Ideally, the resulting scheme $\pi$ should be positive rate, computationally efficient, and achieve optimal error resilience. While interactive coding over large alphabets is well understood, the situation over the binary alphabet has remained evasive. At the present moment, the known schemes over the binary alphabet that achieve a higher error resilience than a trivial adaptation of large alphabet schemes are either still suboptimally error resilient [EKS20], or optimally error resilient with exponential communication complexity [GZ22]. In this work, we construct a scheme achieving optimality in all three parameters: our protocol is positive rate, computationally efficient, and resilient to the optimal $\frac16 - \epsilon$ adversarial errors. Our protocol employs a new type of code that we call a layered code, which may be of independent interest. Like a tree code, a layered code allows the coder to encode a message in an online fashion, but is defined on a graph instead of a tree.