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 a received 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 received word (assuming the received word is within some tolerated corruption rate \(\delta\)). Efficiency of the decoding is quantified by the code’s query complexity \(r\), and decoding 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 [12].
- Quantum locally recoverable code (QLRC)— There are quantum counterparts of LDCs, but they can be transformed into (classical) LDCs which can be decoded well on average [13].
- Expander code— Expander codes are locally decodable provided that the inner code satisfies certain properties; there exist code families with rate approaching one [14].
- Locally correctable code (LCC)— Any family of LCCs can be converted to a family of LDCs whose rate differs by a constant [15]; see [10; Sec. 2.4.1].
- Batch code— Batch codes and LDCs are related [16,17][11; Ch. 10.3].
- Private information retrieval (PIR) code— Any smooth LDC yields a PIR scheme [18]; see also Ref. [19].
- \(q\)-ary linear LCC— Linear LCCs can be converted into LDCs with the same locality \(r\) [10; Sec. 2.4.1].
Primary Hierarchy
Parents
Locally decodable code (LDC)
References
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- S. Yekhanin, Locally Decodable Codes and Private Information Retrieval Schemes (Springer Berlin Heidelberg, 2010) DOI
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2023-05-05)
Cite as:
“Locally decodable code (LDC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/ldc, arXiv:2606.11484