Error Correction by Structural Simplicity: Correcting Samplable Additive Errors

TL;DR

This paper investigates the potential and limits of correcting errors based on their structural simplicity in samplable additive channels, revealing that certain simple errors are uncorrectable while identifying conditions for successful correction.

Contribution

It introduces the concept of samplable additive errors, analyzes their properties, and establishes both limitations and conditions for error correction in such channels.

Findings

Samplable errors with entropy $n^{\

,

are pseudorandom and uncorrectable by efficient codes.

Abstract

This paper explores the possibilities and limitations of error correction by the structural simplicity of error mechanisms. Specifically, we consider channel models, called \emph{samplable additive channels}, in which (a) errors are efficiently sampled without the knowledge of the coding scheme or the transmitted codeword; (b) the entropy of the error distribution is bounded; and (c) the number of errors introduced by the channel is unbounded. For the channels, several negative and positive results are provided. Assuming the existence of one-way functions, there are samplable additive errors of entropy $n^{\epsilon}$ for $\epsilon \in (0,1)$ that are pseudorandom, and thus not correctable by efficient coding schemes. It is shown that there is an oracle algorithm that induces a samplable distribution over $\{0,1\}^n$ of entropy $m = \omega( \log n)$ that is not pseudorandom, but is uncorrectable by efficient schemes of rate less than $1 - m/n - o(1)$. The results indicate that restricting error mechanisms to be efficiently samplable and not pseudorandom is insufficient for error correction. As positive results, some conditions are provided under which efficient error correction is possible.