Editing BCH code (section)

{{short description|Error correction code}}
In [[coding theory]], the '''Bose&ndash;Chaudhuri&ndash;Hocquenghem codes''' ('''BCH codes''') form a class of [[cyclic code|cyclic]] [[Error correction code|error-correcting codes]] that are constructed using [[polynomial]]s over a [[finite field]] (also called a ''[[Finite field|Galois field]]''). BCH codes were invented in 1959 by French mathematician [[Alexis Hocquenghem]], and independently in 1960 by [[Raj Chandra Bose]] and [[D. K. Ray-Chaudhuri]].<ref>{{Harvnb|Reed|Chen|1999|p=189}}</ref><ref>{{harvnb|Hocquenghem|1959}}</ref><ref>{{harvnb|Bose|Ray-Chaudhuri|1960}}</ref> The name ''Bose&ndash;Chaudhuri&ndash;Hocquenghem'' (and the acronym ''BCH'') arises from the initials of the inventors' surnames (mistakenly, in the case of Ray-Chaudhuri).

One of the key features of BCH codes is that during code design, there is a precise control over the number of symbol errors correctable by the code. In particular, it is possible to design binary BCH codes that can correct multiple bit errors. Another advantage of BCH codes is the ease with which they can be decoded, namely, via an [[Abstract algebra|algebraic]] method known as [[syndrome decoding]]. This simplifies the design of the decoder for these codes, using small low-power electronic hardware.

BCH codes are used in applications such as satellite communications,<ref>{{cite web|title=Phobos Lander Coding System: Software and Analysis|url=http://ipnpr.jpl.nasa.gov/progress_report/42-94/94V.PDF |archive-url=https://ghostarchive.org/archive/20221009/http://ipnpr.jpl.nasa.gov/progress_report/42-94/94V.PDF |archive-date=2022-10-09 |url-status=live|access-date=25 February 2012}}</ref> [[compact disc]] players, [[DVD]]s, [[Disk storage|disk drives]], [[USB flash drive]]s, [[solid-state drive]]s,<ref>{{cite book|chapter=BCH Codes for Solid-State-Drives|doi=10.1007/978-981-13-0599-3_11 |chapter-url=https://link.springer.com/chapter/10.1007/978-981-13-0599-3_11|access-date=23 September 2023 |title=Inside Solid State Drives (SSDS) |series=Springer Series in Advanced Microelectronics |date=2018 |last1=Marelli |first1=Alessia |last2=Micheloni |first2=Rino |volume=37 |pages=369–406 |isbn=978-981-13-0598-6 }}</ref> and [[Barcode|two-dimensional bar codes]].