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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]
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.
LDPC codes are now used in many recent high-speed communication standards, such as DVB-S2 (Digital Video Broadcasting – Satellite – Second Generation), WiMAX (IEEE 802.16e standard for microwave communications), High-Speed Wireless LAN (IEEE 802.11n), [14] 10GBase-T Ethernet (802.3an) and G.hn/G.9960 (ITU-T Standard for networking over ...
Convolutional codes are processed on a bit-by-bit basis. They are particularly suitable for implementation in hardware, and the Viterbi decoder allows optimal decoding. Block codes are processed on a block-by-block basis. Early examples of block codes are repetition codes, Hamming codes and multidimensional parity-check codes.
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.
One such type of Gray code is the n-ary Gray code, also known as a non-Boolean Gray code. As the name implies, this type of Gray code uses non-Boolean values in its encodings. For example, a 3-ary Gray code would use the values 0,1,2. [31] The (n, k)-Gray code is the n-ary Gray code with k digits. [63]
The Reed–Solomon code is actually a family of codes, where every code is characterised by three parameters: an alphabet size , a block length, and a message length, with <. The set of alphabet symbols is interpreted as the finite field F {\displaystyle F} of order q {\displaystyle q} , and thus, q {\displaystyle q} must be a prime power .
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 } .