Error-correcting code (ECC)


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 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 \(x\in[1\cdots K]\), \(d(E(e(x)))=x\) [1].


Capacity-achieving Guessing Random Additive Noise Decoding (GRAND) [2].


The modern theory of error-correcting codes is rooted in the foundational work of C. Shannon [3], but error-correcting codes have been used prior to that work [4].




D. Gottesman. Surviving as a quantum computer in a classical world
K. R. Duffy, J. Li, and M. Medard, “Capacity-Achieving Guessing Random Additive Noise Decoding”, IEEE Transactions on Information Theory 65, 4023 (2019). DOI; 1802.07010
C. E. Shannon, “A Mathematical Theory of Communication”, Bell System Technical Journal 27, 379 (1948). DOI
A. Barg, “At the Dawn of the Theory of Codes”, The Mathematical Intelligencer 15, 20 (1993). DOI
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“Error-correcting code (ECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_ecc, title={Error-correcting code (ECC)}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Error-correcting code (ECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.