Finite-dimensional error-correcting code (ECC)[1]

Description

A code is a subset of a set or alphabet, with each element called a codeword. An error-correcting code consists of \(K\) codewords over an alphabet with \(N\) elements such that it is possible to recover the codewords from errors \(E\) from some error set \(\mathcal{E}\).

A common family of codes are the block codes, intended to encode a piece, or block, of a data stream. A block code encodes strings of length \(k\), where each character in the string an element of some fixed alphabet \(\Sigma\), into strings of length \(n\). In other words, a block code encoding is a map from \(\Sigma^k\) to \(\Sigma^n\), where \(N = |\Sigma|^n\), \(K=|\Sigma|^k\), and \(|\Sigma|\) is the number of elements in the alphabet.

Protection

A code corrects errors associated with a noise channel if it is possible to recover any codeword after its coordinates have been changed after going through the channel. More technically, an error-correcting code \((u,\mathcal{E})\) is an encoder function \(u:[1\cdots K]\to[1\cdots N]\) with a set of correctable errors \(E:[1\cdots N]\to [1\cdots M]\) with the following property: there exists a decoder function \(d:[1\cdots M]\to [1\cdots K]\) such that for all \(E\in\cal{E}\) and states \(x\in[1\cdots K]\), \(d(E(e(x)))=x\) [2].

Decoding

Capacity-achieving Guessing Random Additive Noise Decoding (GRAND) [3] (see also [4]).

Notes

The modern theory of error-correcting codes is rooted in the foundational work of C. Shannon [1], but error-correcting codes have been used prior to that work [5].

Parent

Children

Cousin

References

[1]
C. E. Shannon, “A Mathematical Theory of Communication”, Bell System Technical Journal 27, 379 (1948) DOI
[2]
D. Gottesman. Surviving as a quantum computer in a classical world
[3]
K. R. Duffy, J. Li, and M. Medard, “Capacity-Achieving Guessing Random Additive Noise Decoding”, IEEE Transactions on Information Theory 65, 4023 (2019) arXiv:1802.07010 DOI
[4]
K. R. Duffy, J. Li, and M. Medard, “Guessing noise, not code-words”, 2018 IEEE International Symposium on Information Theory (ISIT) (2018) DOI
[5]
A. Barg, “At the Dawn of the Theory of Codes”, The Mathematical Intelligencer 15, 20 (1993) DOI
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Zoo Code ID: ecc_finite

Cite as:
“Finite-dimensional error-correcting code (ECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/ecc_finite
BibTeX:
@incollection{eczoo_ecc_finite, title={Finite-dimensional error-correcting code (ECC)}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/ecc_finite} }
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“Finite-dimensional error-correcting code (ECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/ecc_finite

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/properties/ecc_finite.yml.