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Query-Efficient Locally Decodable Codes of Subexponential Length

Yeow Meng Chee, Tao Feng, San Ling, Huaxiong Wang, Liang Feng Zhang
2011 Computational Complexity  
A k-query locally decodable code (LDC) C : Σ n → Γ N encodes each message x into a codeword C(x) such that each symbol of x can be probabilistically recovered by querying only k coordinates of C(x), even  ...  Yekhanin (in J ACM 55:1-16, 2008) constructed a 3-query LDC of subexponential length, N = exp(exp(O(log n/ log log n))), under the assumption that there are infinitely many Mersenne primes.  ...  Research of Y. M. Chee, S. Ling, and H. Wang is supported in part by the National Research Foundation of Singapore under Research Grant NRF-CRP2-2007-03.  ... 
doi:10.1007/s00037-011-0017-1 fatcat:s3yn4vs3n5fx3fzhbdjv36auhe

Query-Efficient Locally Decodable Codes of Subexponential Length [article]

Yeow Meng Chee, Tao Feng, San Ling, Huaxiong Wang, Liang Feng Zhang
2010 arXiv   pre-print
More precisely, we construct a 3^ r/2-query LDC for every positive integer r<104 and a (3/4)^51· 2^r-query LDC for every integer r≥ 104, both of length N_r, improving the 2^r queries used by Efremenko  ...  We also obtain new efficient private information retrieval (PIR) schemes from the new query-efficient LDCs.  ...  Research of Y. M. Chee, S. Ling, and H. Wang is supported in part by the National Research Foundation of Singapore under Research Grant NRF-CRP2-2007-03.  ... 
arXiv:1008.1617v1 fatcat:2zjmfuez6rcg7okomwimt3oula

Improved Constructions for Query-Efficient Locally Decodable Codes of Subexponential Length

Toshiya ITOH, Yasuhiro SUZUKI
2010 IEICE transactions on information and systems  
The efficiency of a (k,δ,ϵ)-locally decodable code C: F_q^n→ F_q^N is measured by the code length N and the number k of queries.  ...  For any k-query locally decodable code C: F_q^n→ F_q^N, the code length N is conjectured to be exponential of n, however, this was disproved.  ...  Acknowledgments The authors would like to thank Osamu Watanabe for his insightful discussions and valuable comments for the earlier version of the paper.  ... 
doi:10.1587/transinf.e93.d.263 fatcat:24ynywiytvfnbncxwifrzb2cce

Three-Query Locally Decodable Codes with Higher Correctness Require Exponential Length

Anna Gal, Andrew Mills
2012 ACM Transactions on Computation Theory  
A breakthrough result by Yekhanin showed that 3-query linear locally decodable codes may have subexponential length.  ...  Previously, there were no larger than quadratic lower bounds known for locally decodable codes with more than 2 queries, even in the case of 3-query linear codes.  ...  Building on these works, Efremenko [9] gave a construction of subexponential length 3-query linear locally decodable codes without any unproven assumptions.  ... 
doi:10.1145/2077336.2077338 fatcat:angmaledcvbmhmlqiyhpnfxkoi

Towards 3-query locally decodable codes of subexponential length

Sergey Yekhanin
2007 Proceedings of the thirty-ninth annual ACM symposium on Theory of computing - STOC '07  
Towards 3-query locally decodable codes of subexponential length.  ...  Under this conjecture, our constructions yield three query locally decodable codes of length N = exp(n O( 1 log log n ) ) for infinitely many n.  ...  I am indebted to Madhu Sudan for expressing his optimism regarding viability of the approach taken in this paper at an early stage of the work.  ... 
doi:10.1145/1250790.1250830 dblp:conf/stoc/Yekhanin07 fatcat:dh2poknwj5ab3esho6fkjtmhou

Towards 3-query locally decodable codes of subexponential length

Sergey Yekhanin
2008 Journal of the ACM  
Towards 3-query locally decodable codes of subexponential length.  ...  Under this conjecture, our constructions yield three query locally decodable codes of length N = exp(n O( 1 log log n ) ) for infinitely many n.  ...  I am indebted to Madhu Sudan for expressing his optimism regarding viability of the approach taken in this paper at an early stage of the work.  ... 
doi:10.1145/1326554.1326555 fatcat:jr6fpywpgnethkfyzbnhrz3n6e

Local List-Decoding and Testing of Random Linear Codes from High Error

Swastik Kopparty, Shubhangi Saraf
2013 SIAM journal on computing (Print)  
Our main result is that random sparse linear codes are locally testable and locally list-decodable in the high-error regime with only a constant number of queries.  ...  In this paper, we give surprisingly efficient algorithms for list-decoding and testing random linear codes.  ...  Acknowledgements We are very grateful to Madhu Sudan for suggesting these problems to us, for constant encouragement, valuable discussions and for conjecturing the impossibility of our results.  ... 
doi:10.1137/100811945 fatcat:lmqjhkyhuraijbgnkdnronam2y

Local list-decoding and testing of random linear codes from high error

Swastik Kopparty, Shubhangi Saraf
2010 Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10  
Our main result is that random sparse linear codes are locally testable and locally list-decodable in the high-error regime with only a constant number of queries.  ...  In this paper, we give surprisingly efficient algorithms for list-decoding and testing random linear codes.  ...  Acknowledgements We are very grateful to Madhu Sudan for suggesting these problems to us, for constant encouragement, valuable discussions and for conjecturing the impossibility of our results.  ... 
doi:10.1145/1806689.1806748 dblp:conf/stoc/KoppartyS10 fatcat:hpl7g5nyonaetagnppiumddhgy

Locally Decodable Codes with Randomized Encoding [article]

Kuan Cheng, Xin Li, Yu Zheng
2020 arXiv   pre-print
We initiate a study of locally decodable codes with randomized encoding.  ...  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  ...  For locally decodable codes, the rate also depends on the number of queries for decoding.  ... 
arXiv:2001.03692v1 fatcat:5i2gq74l5nc2zbufpzkzwh4zja

Local correctability of expander codes

Brett Hemenway, Rafail Ostrovsky, Mary Wootters
2015 Information and Computation  
In this work, we present the first local-decoding algorithm for expander codes.  ...  Instantiating our construction with inner codes based on finite geometries, we obtain novel locally decodable codes with rate approaching one.  ...  The study of locally decodable codes focuses on the trade-off between rate (the ratio of message length to codeword length) and query complexity (the number of queries made by the decoder).  ... 
doi:10.1016/j.ic.2014.12.013 fatcat:4kwz46gddra3pgwipxbdh4nr4e

Local Correctability of Expander Codes [chapter]

Brett Hemenway, Rafail Ostrovsky, Mary Wootters
2013 Lecture Notes in Computer Science  
In this work, we present the first local-decoding algorithm for expander codes.  ...  Instantiating our construction with inner codes based on finite geometries, we obtain novel locally decodable codes with rate approaching one.  ...  The study of locally decodable codes focuses on the trade-off between rate (the ratio of message length to codeword length) and query complexity (the number of queries made by the decoder).  ... 
doi:10.1007/978-3-642-39206-1_46 fatcat:t4jsddzw3bggrldnizz3aecxkq

Local Correctability of Expander Codes [article]

Brett Hemenway, Rafail Ostrovsky, Mary Wootters
2015 arXiv   pre-print
In this work, we present the first local-decoding algorithm for expander codes.  ...  Instantiating our construction with inner codes based on finite geometries, we obtain novel locally decodable codes with rate approaching one.  ...  The study of locally decodable codes focuses on the trade-off between rate (the ratio of message length to codeword length) and query complexity (the number of queries made by the decoder).  ... 
arXiv:1304.8129v2 fatcat:3pt6ciaakbd5lboriguoaskxiy

Pseudorandom Generators [chapter]

Oded Goldreich
1999 Modern Cryptography, Probabilistic Proofs and Pseudorandomness  
Let A(x; r) be a BPP algorithm that on inputs x of length n, can be simulated by Boolean circuits of size at most most n c , using coin tosses r.  ...  , the efficiency requirement refers to uniform algorithms.  ...  Constructions of 3-query locally decodable codes with subexponential blocklength were recently given by Yekhanin [ Techniques for Hardness Amplification (namely, the Direct Product Theorem and XOR Lemma  ... 
doi:10.1007/978-3-662-12521-2_3 fatcat:d44aduuufjgijhehp4toyijjz4

Pseudorandom Generators [chapter]

Evangelos Kranakis
1986 Primality and Cryptography  
Let A(x; r) be a BPP algorithm that on inputs x of length n, can be simulated by Boolean circuits of size at most most n c , using coin tosses r.  ...  , the efficiency requirement refers to uniform algorithms.  ...  Constructions of 3-query locally decodable codes with subexponential blocklength were recently given by Yekhanin [ Techniques for Hardness Amplification (namely, the Direct Product Theorem and XOR Lemma  ... 
doi:10.1007/978-3-322-96647-6_4 fatcat:6febgpanvrfpbbihvvwqsn77zq

Erasures vs. Errors in Local Decoding and Property Testing

Sofya Raskhodnikova, Noga Ron-Zewi, Nithin Varma, Michael Wagner
2018 Innovations in Theoretical Computer Science  
We use this result to exhibit a property which is testable with a number of queries independent of the length of the input in the presence of erasures, but requires a number of queries that depends on  ...  We prove an analog of a famous result of Goldreich and Levin on local list decodability of the Hadamard code.  ...  Some Applications of Coding Theory in Computational Complexity. CoRR, cs.CC/0409044, 2004. arXiv:cs/0409044. 40 Sergey Yekhanin. Towards 3-query locally decodable codes of subexponential length.  ... 
doi:10.4230/lipics.itcs.2019.63 dblp:conf/innovations/RaskhodnikovaRV19 fatcat:bfbm2dpuvfcb5gi6ddl6gfrbdi
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