# 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

## Decoding

## Notes

## 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

## 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.