Locally decodable code (LDC)[1]
Description
Recall that a block code encodes a length-\(k\) message into a length-\(n\) codeword, which is then sent through a noise channel to yield an error word. Informally, an LDC is a block code for which one can recover any coordinate of the message from at most \(r\) coordinates of the error word (assuming the error word is within some tolerated corruption rate \(\delta\)). Efficiency of the correction is quantified by the code's query complexity \(r\), and correction is performed by sampling subsets of \(r\) bits.
LDCs have applications in computational complexity theory and cryptography [3–5][2; Sec. 17.4].
Modified versions of local decodability include relaxed local decodability [6].
Rate
Families of LDCs with query complexity \(r=2\) need \(n\) to scale exponentially with \(k\) [7,8].Decoding
LDCs admit decoders whose runtime scales polylogarithmically with \(n\).Cousins
- Locally testable code (LTC)— There are relations between LDCs and LTCs [11].
- Quantum locally recoverable code (QLRC)— There are quantum versions of LDCs, but they can be transformed into LDCs which can be decoded well on average [12].
- Expander code— Expander codes are locally decodable provided that the inner code satisfies certain properties; there exist code families with rate approaching one [13].
- Locally correctable code (LCC)— Any family of LCCs can be converted to a family of LDCs whose rate differs by a constant [14]; see [9; Sec. 2.4.1].
- Batch code— Batch codes and LDCs are related [15,16][10; Ch. 10.3].
- Private information retrieval (PIR) code— Any smooth LDC yields a PIR scheme [17]; see also Ref. [18].
Primary Hierarchy
Parents
Locally decodable code (LDC)
Children
Linear LCCs can be converted into LDCs with the same locality \(r\) [9; Sec. 2.4.1].
References
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- A. Ben-Aroya, O. Regev, and R. de Wolf, “A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs”, 2008 49th Annual IEEE Symposium on Foundations of Computer Science (2008) arXiv:0705.3806 DOI
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- O. Goldreich, H. Karloff, L. J. Schulman, and L. Trevisan, “Lower bounds for linear locally decodable codes and private information retrieval”, Proceedings 17th IEEE Annual Conference on Computational Complexity 175 DOI
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- Gopi, Sivakanth. Locality in coding theory. Diss. Princeton University, 2018.
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- I. F. Blake, Essays on Coding Theory (Cambridge University Press, 2024) DOI
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- [12]
- J. Briët and R. de Wolf, “Locally Decodable Quantum Codes”, (2008) arXiv:0806.2101
- [13]
- B. Hemenway, R. Ostrovsky, and M. Wootters, “Local Correctability of Expander Codes”, (2015) arXiv:1304.8129
- [14]
- A. Bhattacharyya, S. Gopi, and A. Tal, “Lower bounds for 2-query LCCs over large alphabet”, (2017) arXiv:1611.06980
- [15]
- Y. Ishai, E. Kushilevitz, R. Ostrovsky, and A. Sahai, “Batch codes and their applications”, Proceedings of the thirty-sixth annual ACM symposium on Theory of computing (2004) DOI
- [16]
- R. Henry, “Polynomial Batch Codes for Efficient IT-PIR”, Proceedings on Privacy Enhancing Technologies 2016, 202 (2016) DOI
- [17]
- H. Zhang, E. Yaakobi, and N. Silberstein, “Multiset Combinatorial Batch Codes”, (2017) arXiv:1701.02708
- [18]
- S. Yekhanin, Locally Decodable Codes and Private Information Retrieval Schemes (Springer Berlin Heidelberg, 2010) DOI
Page edit log
- Victor V. Albert (2023-05-05) — most recent
Cite as:
“Locally decodable code (LDC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/ldc