Error Correction for Message Streams

TL;DR

This paper introduces an encoding and decoding scheme enabling low-space decoders to compute any function of the message with high error resilience in streaming settings.

Contribution

It presents a novel encoding and decoding method allowing low-space streaming decoders to compute arbitrary functions despite a constant fraction of errors.

Findings

Decoding in low space is possible for any function with error fraction less than 1/4.

The scheme achieves polylogarithmic overhead in space for decoding.

It extends the capabilities of local decoding to arbitrary function computation.

Abstract

In the setting of error correcting codes, Alice wants to send a message $x \in \{0,1\}^n$ to Bob via an encoding $\text{enc}(x)$ that is resilient to error. In this work, we investigate the scenario where Bob is a low space decoder. More precisely, he receives Alice's encoding $\text{enc}(x)$ bit-by-bit and desires to compute some function $f(x)$ in low space. A generic error-correcting code does not accomplish this because decoding is a very global process and requires at least linear space. Locally decodable codes partially solve this problem as they allow Bob to learn a given bit of $x$ in low space, but not compute a generic function $f$. Our main result is an encoding and decoding procedure where Bob is still able to compute any such function $f$ in low space when a constant fraction of the stream is corrupted. More precisely, we describe an encoding function $\text{enc}(x)$ of length $\text{poly}(n)$ so that for any decoder (streaming algorithm) $A$ that on input $x$ computes $f(x)$ in space $s$, there is an explicit decoder $B$ that computes $f(x)$ in space $s \cdot \text{polylog}(n)$ as long as there were not more than $\frac14 - \varepsilon$ fraction of (adversarial) errors in the input stream $\text{enc}(x)$.