# High-rate locally-correctable and locally-testable codes with sub-polynomial query complexity

@article{Kopparty2015HighrateLA, title={High-rate locally-correctable and locally-testable codes with sub-polynomial query complexity}, author={Swastik Kopparty and Or Meir and Noga Ron-Zewi and Shubhangi Saraf}, journal={Proceedings of the forty-eighth annual ACM symposium on Theory of Computing}, year={2015} }

In this work, we construct the first locally-correctable codes (LCCs), and locally-testable codes (LTCs) with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist LCCs and LTCs with block length n, constant rate (which can even be taken arbitrarily close to 1) and constant relative distance, whose query complexity is exp(Õ(√logn)) (for LCCs) and (logn)O(loglogn) (for LTCs). Previously such codes were known to exist only with Ω(n…

## 68 Citations

### High-Rate Locally Correctable and Locally Testable Codes with Sub-Polynomial Query Complexity

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This work studies RLDCs in the standard Hamming-error setting, and introduces their variants in the insertion and deletion (Insdel) error setting, which are further motivated by recent advances in DNA random access bio-technologies.

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It is shown that PRM codes are shorter than generalized Reed-Muller~(GRM) codes in LCCs, while maintaining the same values on the query complexities and the message lengths.

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This work constructs the first LCCs and LTCs with constant rate, constant relative distance, and sub-polynomial query complexity, and approaches the Singleton bound over large (but constant size) alphabet.

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