Local Correctability of Expander Codes

TL;DR

This paper introduces the first local-decoding algorithm for expander codes, enabling high-probability symbol recovery with sublinear reads, and constructs new high-rate locally decodable codes using finite geometry-based inner codes.

Contribution

It presents a novel local-decoding algorithm for expander codes, linking smooth inner codes to efficient local correction in noisy environments, and constructs high-rate codes approaching one.

Findings

Achieved local correction with high probability by reading $N^ extepsilon$ symbols.

Constructed high-rate locally decodable codes using finite geometries.

Provided an alternative to existing multiplicity and lifted codes.

Abstract

In this work, we present the first local-decoding algorithm for expander codes. This yields a new family of constant-rate codes that can recover from a constant fraction of errors in the codeword symbols, and where any symbol of the codeword can be recovered with high probability by reading $N^\epsilon$ symbols from the corrupted codeword, where $N$ is the block-length of the code. Expander codes, introduced by Sipser and Spielman, are formed from an expander graph $G = (V,E)$ of degree $d$, and an inner code of block-length $d$ over an alphabet $\Sigma$. Each edge of the expander graph is associated with a symbol in $\Sigma$. A string in $\Sigma^{E}$ will be a codeword if for each vertex in $V$, the symbols on the adjacent edges form a codeword in the inner code. We show that if the inner code has a smooth reconstruction algorithm in the noiseless setting, then the corresponding expander code has an efficient local-correction algorithm in the noisy setting. Instantiating our construction with inner codes based on finite geometries, we obtain novel locally decodable codes with rate approaching one. This provides an alternative to the multiplicity codes of Kopparty, Saraf and Yekhanin (STOC '11) and the lifted codes of Guo, Kopparty and Sudan (ITCS '13).