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


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 \(x\in[1\cdots K]\), \(d(E(e(x)))=x\) [2].


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


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].Boolean networks, designed to model gene regulatory networks, generically develop error-correcting codes when they are evolved to perform computations [6].





C. E. Shannon, “A Mathematical Theory of Communication”, Bell System Technical Journal 27, 379 (1948) DOI
D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
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
K. R. Duffy, J. Li, and M. Medard, “Guessing noise, not code-words”, 2018 IEEE International Symposium on Information Theory (ISIT) (2018) DOI
A. Barg, “At the Dawn of the Theory of Codes”, The Mathematical Intelligencer 15, 20 (1993) DOI
T. McCourt, I. R. Fiete, and I. L. Chuang, “Noisy dynamical systems evolve error correcting codes and modularity”, (2023) arXiv:2303.14448
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Zoo Code ID: ecc_finite

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“Finite-dimensional error-correcting code (ECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@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={} }
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“Finite-dimensional error-correcting code (ECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.