# 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 \(\mathcal{E}\).

## 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

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

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## 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
- [6]
- T. McCourt, I. R. Fiete, and I. L. Chuang, “Noisy dynamical systems evolve error correcting codes and modularity”, (2023) arXiv:2303.14448

## Page edit log

- Victor V. Albert (2022-03-22) — most recent
- Victor V. Albert (2021-11-24)

## 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

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