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The Hamming(7,4) code may be written as a cyclic code over GF(2) with generator + +. In fact, any binary Hamming code of the form Ham(r, 2) is equivalent to a cyclic code, [3] and any Hamming code of the form Ham(r,q) with r and q-1 relatively prime is also equivalent to a cyclic code. [4]
The generator polynomial of the BCH code is defined as the least common multiple g(x) = lcm(m 1 (x),…,m d − 1 (x)). It can be seen that g(x) is a polynomial with coefficients in GF(q) and divides x n − 1. Therefore, the polynomial code defined by g(x) is a cyclic code.
However cyclic codes can indeed detect most bursts of length >. The reason is that detection fails only when the burst is divisible by g ( x ) {\displaystyle g(x)} . Over binary alphabets, there exist 2 ℓ − 2 {\displaystyle 2^{\ell -2}} bursts of length ℓ {\displaystyle \ell } .
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.
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:
These examples also belong to the class of linear codes, and hence they are called linear block codes. More particularly, these codes are known as algebraic block codes, or cyclic block codes, because they can be generated using Boolean polynomials. Algebraic block codes are typically hard-decoded using algebraic decoders. [jargon]
Being a code that achieves this optimal trade-off, the Reed–Solomon code belongs to the class of maximum distance separable codes. While the number of different polynomials of degree less than k and the number of different messages are both equal to q k {\displaystyle q^{k}} , and thus every message can be uniquely mapped to such a polynomial ...
Cyclic codes are a kind of block code with the property that the circular shift of a codeword will always yield another codeword. This motivates the following general definition: For a string s over an alphabet Σ , let shift ( s ) denote the set of circular shifts of s , and for a set L of strings, let shift ( L ) denote the set of all ...