# BCH code

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In coding theory the BCH codes form a class of parameterised error-correcting codes which have been the subject of much academic attention in the last fifty years. BCH codes were invented in 1959 by Hocquenghem, and independently in 1960 by Bose and Ray-Chaudhuri [1]. The acronym BCH comprises the initials of these inventors' names.

The principal advantage of BCH codes is the ease with which they can be decoded, via an elegant algebraic method known as syndrome decoding. This allows very simple electronic hardware to perform the task, obviating the need for a computer, and meaning that a decoding device may be made small and low-powered. As a class of codes, they are also highly flexible, allowing control over block length and acceptable error thresholds, meaning that a custom code can be designed to a given specification (subject to mathematical constraints).

Reedâ€“Solomon codes, which are BCH codes, are used in applications such as satellite communications, compact disc players, DVDs, disk drives, and two-dimensional bar codes.

In technical terms a BCH code is a multilevel cyclic variable-length digital error-correcting code used to correct multiple random error patterns. BCH codes may also be used with multilevel phase-shift keying whenever the number of levels is a prime number or a power of a prime number. A BCH code in 11 levels has been used to represent the 10 decimal digits plus a sign digit.[2]

BCH codes are also useful in theoretical computer science, for instance in the MAXEkSAT problem.

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### Construction

A BCH code is a polynomial code over a finite field with a particularly chosen generator polynomial. It is also a cyclic code.