Some Applications of Coding Theory in Computational Complexity

TL;DR

This survey explores how error-correcting codes, especially locally-testable and locally-decodable codes, are applied in computational complexity and cryptography, highlighting their roles in probabilistically checkable proofs and cryptographic protocols.

Contribution

It provides a comprehensive overview of the applications of coding theory in complexity and cryptography, emphasizing recent developments and connections.

Findings

Locally decodable codes enable sub-linear time error correction and are linked to private information retrieval.

Locally testable codes facilitate sub-linear time error detection and are central to probabilistically checkable proofs.

These codes have significant implications for average-case complexity and cryptographic security.

Abstract

Error-correcting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locally-testable and locally-decodable error-correcting codes, and their applications to complexity theory and to cryptography. Locally decodable codes are error-correcting codes with sub-linear time error-correcting algorithms. They are related to private information retrieval (a type of cryptographic protocol), and they are used in average-case complexity and to construct ``hard-core predicates'' for one-way permutations. Locally testable codes are error-correcting codes with sub-linear time error-detection algorithms, and they are the combinatorial core of probabilistically checkable proofs.