128x18 Error Correction Code
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They were conceived in 1966 by Dave Forney as a solution to the problem of finding a code that has both exponentially decreasing error probability with increasing block length and polynomial-time decoding
Concatenated Codes Example
complexity.[1] Concatenated codes became widely used in space communications in the 1970s. Contents 1 error correcting code hamming Background 2 Description 3 Properties 4 Decoding concatenated codes 4.1 Remarks 5 Applications 6 Turbo codes: A parallel concatenation approach
Error Correcting Code Example
7 See also 8 References 9 Further reading 10 External links Background[edit] The field of channel coding is concerned with sending a stream of data at the highest possible rate over a given communications concatenated codes pdf channel, and then decoding the original data reliably at the receiver, using encoding and decoding algorithms that are feasible to implement in a given technology. Shannon's channel coding theorem shows that over many common channels there exist channel coding schemes that are able to transmit data reliably at all rates R {\displaystyle R} less than a certain threshold C {\displaystyle C} , called the channel capacity of error correcting codes pdf the given channel. In fact, the probability of decoding error can be made to decrease exponentially as the block length N {\displaystyle N} of the coding scheme goes to infinity. However, the complexity of a naive optimum decoding scheme that simply computes the likelihood of every possible transmitted codeword increases exponentially with N {\displaystyle N} , so such an optimum decoder rapidly becomes infeasible. In his doctoral thesis, Dave Forney showed that concatenated codes could be used to achieve exponentially decreasing error probabilities at all data rates less than capacity, with decoding complexity that increases only polynomially with the code block length. Description[edit] Schematic depiction of a concatenated code built upon an inner code and an outer code. This is a pictorial representation of a code concatenation, and, in particular, the Reed–Solomon code with n=q=4 and k=2 is used as the outer code and the Hadamard code with n=q and k=log q is used as the inner code. Overall, the concatenated code is a [ q 2 , k log q ] {\displaystyle [q^{2},k\log q]} -code. Let Cin be a [n, k, d] code, that is, a block code of length n, dimension k, minimum Hamming distance d, and rate
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