Locally Decodable Codes with Randomized Encoding

TL;DR

This paper explores the use of randomized encoding in locally decodable codes to improve the tradeoff between code rate and query complexity, applicable to both error models, with potential for more efficient data recovery.

Contribution

It introduces a novel approach using randomized encoding to enhance rate-query tradeoffs in locally decodable codes across various error models.

Findings

Achieves better rate-query tradeoff with randomized encoding.

Works for both Hamming and edit error models.

Improves efficiency of local decoding processes.

Abstract

We initiate a study of locally decodable codes with randomized encoding. Standard locally decodable codes are error correcting codes with a deterministic encoding function and a randomized decoding function, such that any desired message bit can be recovered with good probability by querying only a small number of positions in the corrupted codeword. This allows one to recover any message bit very efficiently in sub-linear or even logarithmic time. Besides this straightforward application, locally decodable codes have also found many other applications such as private information retrieval, secure multiparty computation, and average-case complexity. However, despite extensive research, the tradeoff between the rate of the code and the number of queries is somewhat disappointing. For example, the best known constructions still need super-polynomially long codeword length even with a logarithmic number of queries, and need a polynomial number of queries to achieve a constant rate. In this paper, we show that by using a randomized encoding, in several models we can achieve significantly better rate-query tradeoff. In addition, our codes work for both the standard Hamming errors, and the more general and harder edit errors.