BCH code
BCH codes are a class of cyclic error-correcting codes widely used in digital communication and data storage systems. The acronym BCH stands for Bose-Chaudhuri-Hocquenghem, named after Raj Chandra Bose, Dwijendra Kumar Ray-Chaudhuri, and Alexis Hocquenghem, who independently discovered them in 1959 and 1957, respectively.
BCH codes are defined over finite fields (Galois fields), denoted as GF(q), where q is a prime power. A common special case is when q = 2, resulting in binary BCH codes. These codes are powerful because they allow for precise control over the number of errors they can correct.
A BCH code is characterized by two parameters: n, the block length of the code, and t, the number of errors the code is guaranteed to correct. The code is designed to correct up to t errors within a block of n symbols. The code's design relies on specifying a generator polynomial, g(x), which is a factor of xn - 1 over the finite field GF(q). The roots of g(x) determine the error-correcting capability of the code. Specifically, the roots are consecutive powers of a primitive element of an extension field of GF(q).
Encoding a message using a BCH code involves transforming the message into a codeword such that the codeword is divisible by the generator polynomial g(x). This is typically accomplished through polynomial division or using a generator matrix derived from g(x).
Decoding a BCH code involves using algebraic decoding techniques to identify and correct errors in the received codeword. These techniques often involve calculating syndromes (remainders of the received word when divided by the generator polynomial) and using the syndrome values to determine the locations and values of the errors. Algorithms like the Peterson-Gorenstein-Zierler (PGZ) algorithm or the Berlekamp-Massey algorithm are commonly employed for BCH decoding.
Reed-Solomon codes are a special type of non-binary BCH code that are particularly effective at correcting burst errors (multiple consecutive errors). They are defined over GF(2m) for some positive integer m.
BCH codes find applications in diverse areas, including:
- Data storage: Hard disk drives, solid-state drives (SSDs), and optical discs (CDs, DVDs, Blu-rays) utilize BCH codes for error correction.
- Wireless communication: BCH codes are used in cellular communication, satellite communication, and wireless local area networks (WLANs).
- Digital television: BCH codes are used for error correction in digital video broadcasting (DVB) systems.
- Memory systems: BCH codes are implemented in memory systems, such as RAM, to improve data reliability.
The flexibility in choosing code parameters (n and t) and the availability of relatively efficient decoding algorithms have contributed to the widespread adoption of BCH codes in various applications.