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A cyclic redundancy check (CRC) is an error-detecting code commonly used in digital networks and storage devices to detect accidental changes to digital data. [1][2] Blocks of data entering these systems get a short check value attached, based on the remainder of a polynomial division of their contents. On retrieval, the calculation is repeated ...
Burst error-correcting code. In coding theory, burst error-correcting codes employ methods of correcting burst errors, which are errors that occur in many consecutive bits rather than occurring in bits independently of each other. Many codes have been designed to correct random errors.
Mathematics of cyclic redundancy checks. The cyclic redundancy check (CRC) is based on division in the ring of polynomials over the finite field GF (2) (the integers modulo 2), that is, the set of polynomials where each coefficient is either zero or one, and arithmetic operations wrap around. Any string of bits can be interpreted as the ...
Turbo code; Walsh–Hadamard code; Cyclic redundancy checks (CRCs) can correct 1-bit errors for messages at most bits long for optimal generator polynomials of degree , see Mathematics of cyclic redundancy checks § Bitfilters
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.
A CRC has properties that make it well suited for detecting burst errors. CRCs are particularly easy to implement in hardware and are therefore commonly used in computer networks and storage devices such as hard disk drives. The parity bit can be seen as a special-case 1-bit CRC.
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.
It is based on a Markov chain with two states G (for good or gap) and B (for bad or burst). In state G the probability of transmitting a bit correctly is k and in state B it is h. Usually, [4] it is assumed that k = 1.