Local Correctability of Expander Codes (1304.8129v2)

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).

• The paper introduces the first local-decoding algorithm for expander codes, demonstrating their local correctability relies on inner codes with smooth local reconstruction properties.
• The research shows that using inner codes, such as those based on finite geometries, provides a complementary method for constructing high-rate locally decodable codes with constant rates.
• This work enhances the practical application of expander codes in systems requiring quick localized decoding, like distributed coding or data storage, extending their traditional global properties to local domains.

Local Correctability of Expander Codes

The paper "Local Correctability of Expander Codes" presents a significant advancement in coding theory, particularly in the efficient local decoding of expander codes using local reconstruction techniques. Expander codes, initially introduced by Sipser and Spielman, are known for their efficient decoding capabilities, especially in linear codes. The primary contribution of this paper is to demonstrate the local correctability of expander codes while maintaining a constant rate and expander-like properties.

The authors establish the first local-decoding algorithm for expander codes, extending their usability to instances where errors corrupt the encoded messages. This involves recovering any symbol of a codeword with high probability by examining only a sublinear number of symbols from the corrupted codeword, denoted by $N$ in the document, where $N$ is the block length of the code. The concept is based on a $d$-regular expander graph and inner codes with specific local reconstruction algorithms, ensuring effective local correction from noisy settings.

Key Contributions

• Local-Decoding Algorithm: The paper introduces a novel algorithm that allows local correction of expander codes, which were traditionally decoded globally. This is particularly useful in scenarios where accessing the full data is costly or impractical.
• Smooth Reconstruction Requirement: For the expander code to be locally correctable, the inner code must possess a smooth local reconstruction algorithm. Specifically, the paper proves that if the inner code can be reconstructed smoothly without noise, the corresponding expander code can handle certain noise and be corrected locally.
• Comparison with Existing Constructions: By utilizing inner codes based on finite geometries, the authors propose alternatives to the multiplicity codes and lifted codes, presenting a complementary construction method for locally decodable codes with a rate approaching one.
• Generalization to High-Rate LDCs: The research provides a method to generalize the argument from low-query to high-rate scenarios, demonstrating that codes with efficient local reconstruction can become powerful tools in constructing high-rate locally decodable codes.

Numerical Results

While the paper does not focus explicitly on numerical experimentation, it discusses theoretical bounds on query complexities and failure probabilities of decoding algorithms. The authors demonstrate through theoretical analysis that expander codes derived from specific inner codes maintain constant rates while offering efficient local correction capabilities.

For instance, when $d$ (degree of the expander graph) and the size of the alphabet $\Sigma$ are constants, the correction algorithm can query sublinear portions of the code with probabilistic guarantees of success, given certain expansive properties of the graph are met.

Implications and Future Developments

The implications of this paper are twofold. Practically, it provides an enhanced framework for using expander codes in scenarios requiring quick localized decoding, particularly in distributed coding systems and applications where localized errors are prevalent. Theoretically, it extends the applicability of expander codes, traditionally noted for global properties, into domains relying on local code properties while preserving the codes’ efficiency and resilience attributes.

Future developments might explore greater diversification of inner codes with stronger locality properties or constructing expander graphs with more optimal spectral properties to improve the bounds on error tolerance and query complexity. Additionally, experimenting with hardware implementations might yield insights into real-world performance, taking theoretical advancements to practical applications in communications and data storage.

Overall, by integrating smooth local reconstruction approaches within expander code frameworks, the paper advances the understanding of error-correcting codes, offering a robust technique for both digital communication and data reliability fields.