Long code[1][2]

Description

Locally testable \([[2^{2^k},k,d]]\) code. The encoder maps a \(k\)-bit string into a codeword that consists of the values of all Boolean functions on the \(k\)-bit string. The code is not practical, but is useful for certain probabilistically checkable proof (PCP) constructions [3].

Parent

Child

  • Hadamard code — The Hadamard code is a subcode of the long code and can be obtained by restricting the long-code construction to only linear functions.

References

[1]
J. Håstad, “Some optimal inapproximability results”, Journal of the ACM 48, 798 (2001) DOI
[2]
M. Bellare, O. Goldreich, and M. Sudan, “Free Bits, PCPs, and Nonapproximability---Towards Tight Results”, SIAM Journal on Computing 27, 804 (1998) DOI
[3]
P. Harsha et al., “Limits of Approximation Algorithms: PCPs and Unique Games (DIMACS Tutorial Lecture Notes)”, (2010) arXiv:1002.3864
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Zoo Code ID: long

Cite as:
“Long code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/long
BibTeX:
@incollection{eczoo_long, title={Long code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/long} }
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https://errorcorrectionzoo.org/c/long

Cite as:

“Long code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/long

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/bits/ltc/long.yml.