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Cyclic codes are not only simple to implement but have the benefit of being particularly well suited for the detection of burst errors: contiguous sequences of erroneous data symbols in messages. This is important because burst errors are common transmission errors in many communication channels, including magnetic and optical storage devices.
In coding theory, the Bose–Chaudhuri–Hocquenghem codes (BCH codes) form a class of cyclic error-correcting codes that are constructed using polynomials over a finite field (also called a 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 ...
In fact, cyclic codes can also correct cyclic burst errors along with burst errors. Cyclic burst errors are defined as A cyclic burst of length t {\displaystyle t} is a vector whose nonzero components are among t {\displaystyle t} (cyclically) consecutive components, the first and the last of which are nonzero.
Cyclic redundancy checks (CRCs) can correct 1-bit errors for messages at most ... Error-Correction Coding for Digital Communications. New York, USA: Plenum Press.
Proof. We need to prove that if you add a burst of length to a codeword (i.e. to a polynomial that is divisible by ()), then the result is not going to be a codeword (i.e. the corresponding polynomial is not divisible by ()).
Error-correcting codes are used in lower-layer communication such as cellular network, high-speed fiber-optic communication and Wi-Fi, [11] [12] as well as for reliable storage in media such as flash memory, hard disk and RAM. [13] Error-correcting codes are usually distinguished between convolutional codes and block codes:
So CRC method can be used to correct single-bit errors as well (within those limits, e.g. 32,767 bits with optimal generator polynomials of degree 16). Since all odd errors leave an odd residual, all even an even residual, 1-bit errors and 2-bit errors can be distinguished.
Computation of a cyclic redundancy check is derived from the mathematics of polynomial division, modulo two. In practice, it resembles long division of the binary message string, with a fixed number of zeroes appended, by the "generator polynomial" string except that exclusive or operations replace subtractions.