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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]
Classical (algebraic) block codes and convolutional codes are frequently combined in concatenated coding schemes in which a short constraint-length Viterbi-decoded convolutional code does most of the work and a block code (usually Reed–Solomon) with larger symbol size and block length "mops up" any errors made by the convolutional decoder ...
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.
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 } .
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.
A cyclic AN code is a principal ideal of the ring []. There are integers and where = and , satisfy the definition of an AN code. Cyclic AN codes are a subset of cyclic codes and have the same properties.
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.
Linear block codes have the property of linearity, i.e. the sum of any two codewords is also a code word, and they are applied to the source bits in blocks, hence the name linear block codes. There are block codes that are not linear, but it is difficult to prove that a code is a good one without this property.