Constructing Asymptotically Good Relaxed Locally Correctable Codes with Polylogarithmic Query Complexity

Core Concepts

We construct the first asymptotically good relaxed locally correctable codes with polylogarithmic query complexity, bringing the upper bound polynomially close to the lower bound.

Abstract

The key highlights and insights of the content are:

The authors construct the first asymptotically good relaxed locally correctable codes (RLCCs) with polylogarithmic query complexity. This improves upon the previous best upper bound of (log n)^(O(log log log n)) queries.

The authors use a new operation called "nesting" to boost the block length of an RLCC. Nesting an RLCC inside a high-rate locally testable code (LTC) allows the RLCC to inherit the larger block length while incurring only an additive cost in rate and query complexity.

By iteratively nesting the RLCC in a sequence of LTCs with increasing block lengths, the authors are able to construct an RLCC with arbitrarily large block length, constant rate, and polylogarithmic query complexity.

The key to the construction is using the LTC's local testing algorithm to efficiently "zoom in" on a smaller RLCC within the larger input, rather than having to recursively call the smaller RLCC's corrector multiple times.

The authors also show how to obtain non-explicit RLCCs with polylogarithmic queries that approach the Gilbert-Varshamov bound for rate-distance tradeoffs.

The final RLCC has subconstant correcting radius due to the distance limitations of the high-rate LTCs used. This is remedied by nesting in one final LTC with constant rate and distance.

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Key Insights Distilled From

Relaxed Local Correctability from Local Testing

by Vinayak M. K... at arxiv.org 04-02-2024

https://arxiv.org/pdf/2306.17035.pdf

Deeper Inquiries

What are some potential applications of asymptotically good RLCCs with polylogarithmic query complexity

Asymptotically good RLCCs with polylogarithmic query complexity have various potential applications in the field of error correction coding. One key application is in distributed storage systems, where data is stored across multiple servers or nodes. In such systems, RLCCs can be used to ensure data integrity and reliability by allowing for efficient error correction and detection with minimal communication overhead. Additionally, RLCCs can be applied in cloud computing environments to protect data during transmission and storage, ensuring data security and reliability. Furthermore, RLCCs can be utilized in communication systems, such as wireless networks and satellite communications, to enhance the robustness of data transmission and reception in the presence of noise and errors. Overall, the efficient error correction capabilities of RLCCs make them valuable in a wide range of applications where data integrity and reliability are paramount.

Can the techniques developed in this work be extended to construct locally decodable codes (LDCs) with similar parameters

The techniques developed in this work can potentially be extended to construct locally decodable codes (LDCs) with similar parameters. By leveraging the nesting operation and the concept of locally testable codes (LTCs), it may be possible to design LDCs with asymptotically good properties, such as constant rate and distance, along with polylogarithmic query complexity. The key idea would be to adapt the nesting framework to suit the requirements of LDCs, ensuring that the decoding process can be performed efficiently with a small number of queries. By carefully selecting and combining LTCs with appropriate properties, it may be feasible to achieve similar tradeoffs in LDCs as demonstrated in this work for RLCCs. Further research and exploration in this direction could lead to the development of efficient and high-performance LDCs with polylogarithmic query complexity.

How might the nesting operation be generalized or modified to further improve the rate-query tradeoff for RLCCs

The nesting operation can be generalized or modified in several ways to further improve the rate-query tradeoff for RLCCs. One possible extension is to explore different combinations of codes for nesting, such as using nested codes with varying rates and distances to achieve a more optimal tradeoff. Additionally, researchers could investigate the impact of nesting multiple codes in a hierarchical manner, where codes are nested within codes in a structured hierarchy. This hierarchical nesting approach may offer additional flexibility in balancing rate, distance, and query complexity. Furthermore, the nesting operation could be enhanced by incorporating adaptive strategies that dynamically adjust the nesting process based on the specific properties of the codes involved. By exploring these and other variations of the nesting operation, it may be possible to further optimize the performance of RLCCs and achieve even better tradeoffs between rate and query complexity.