Local Correctability of Expander Codes

by Brett Hemenway, Rafail Ostrovsky, and Mary Wootters

Oded's comments

As stated in the abstract (below), the point is meeting the known result, which was first obtained by a novel construction (i.e., multiplicity codes), by using known constructions.

The original 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 $N^\eps$ 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).

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