Editing Error detection and correction (section)

=== Forward error correction ===
[[Forward error correction]] (FEC) is a process of adding [[redundancy (information theory)|redundant data]] such as an [[error-correcting code]] (ECC) to a message so that it can be recovered by a receiver even when a number of errors (up to the capability of the code being used) are introduced, either during the process of transmission or on storage. Since the receiver does not have to ask the sender for retransmission of the data, a [[backchannel]] is not required in forward error correction. Error-correcting codes are used in [[Physical layer|lower-layer]] communication such as [[cellular network]], high-speed [[fiber-optic communication]] and [[Wi-Fi]],<ref>{{cite book |last1=Shah |first1=Pradeep M. |last2=Vyavahare |first2=Prakash D. |last3=Jain |first3=Anjana |title=2015 Radio and Antenna Days of the Indian Ocean (RADIO) |chapter=Modern error correcting codes for 4G and beyond: Turbo codes and LDPC codes |date=September 2015 |pages=1–2 |doi=10.1109/RADIO.2015.7323369 |isbn=978-9-9903-7339-4 |s2cid=28885076 |chapter-url=https://www.researchgate.net/publication/301611980 |access-date=22 May 2022}}</ref><ref>{{cite journal |title=IEEE SA - IEEE 802.11ac-2013 |journal=IEEE Standards Association |url=https://standards.ieee.org/ieee/802.11ac/4473/ |language=en |access-date=2022-05-22 |archive-date=2022-05-22 |archive-url=https://web.archive.org/web/20220522205452/https://standards.ieee.org/ieee/802.11ac/4473/ |url-status=dead }}</ref> as well as for reliable storage in media such as [[flash memory]], [[hard disk]] and [[ECC memory|RAM]].<ref>{{cite web |title=Transition to Advanced Format 4K Sector Hard Drives {{!}} Seagate US |url=https://www.seagate.com/sg/en/tech-insights/advanced-format-4k-sector-hard-drives-master-ti/ |website=Seagate.com |access-date=22 May 2022 |language=en-us}}</ref>

Error-correcting codes are usually distinguished between [[convolutional code]]s and [[block code]]s:
* ''Convolutional codes'' are processed on a bit-by-bit basis. They are particularly suitable for implementation in hardware, and the [[Viterbi decoder]] allows [[maximum likelihood decoding|optimal decoding]].
* ''Block codes'' are processed on a [[Block (telecommunications)|block-by-block]] basis. Early examples of block codes are [[repetition code]]s, [[Hamming code]]s and [[multidimensional parity-check code]]s. They were followed by a number of efficient codes, [[Reed–Solomon code]]s being the most notable due to their current widespread use. [[Turbo code]]s and [[low-density parity-check code]]s (LDPC) are relatively new constructions that can provide almost [[:Category:Capacity-approaching codes|optimal efficiency]].

[[Shannon's theorem]] is an important theorem in forward error correction, and describes the maximum [[information rate]] at which reliable communication is possible over a channel that has a certain error probability or [[signal-to-noise ratio]] (SNR). This strict upper limit is expressed in terms of the [[channel capacity]]. More specifically, the theorem says that there exist codes such that with increasing encoding length the probability of error on a [[channel model|discrete memoryless channel]] can be made arbitrarily small, provided that the [[code rate]] is smaller than the channel capacity. The code rate is defined as the fraction ''k/n'' of ''k'' source symbols and ''n'' encoded symbols.

The actual maximum code rate allowed depends on the error-correcting code used, and may be lower. This is because Shannon's proof was only of existential nature, and did not show how to construct codes that are both optimal and have [[polynomial time|efficient]] encoding and decoding algorithms.