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The American mathematician Richard Hamming pioneered this field in the 1940s and invented the first error-correcting code in 1950: the Hamming (7,4) code. [ 5 ] FEC can be applied in situations where re-transmissions are costly or impossible, such as one-way communication links or when transmitting to multiple receivers in multicast .
The description above is given for what is now called a serially concatenated code. Turbo codes, as described first in 1993, implemented a parallel concatenation of two convolutional codes, with an interleaver between the two codes and an iterative decoder that passes information forth and back between the codes. [6]
A code has all-symbol locality and availability if every code symbol can be recovered from disjoint repair sets of other symbols, each set of size at most symbols. Such codes are called ( r , t ) a {\displaystyle (r,t)_{a}} -LRC.
The Reed–Muller RM(r, m) code of order r and length N = 2 m is the code generated by v 0 and the wedge products of up to r of the v i, 1 ≤ i ≤ m (where by convention a wedge product of fewer than one vector is the identity for the operation).
For example, the widely used (255,223) code can be converted to a (160,128) code by padding the unused portion of the source block with 95 binary zeroes and not transmitting them. At the decoder, the same portion of the block is loaded locally with binary zeroes. The QR code, Ver 3 (29×29) uses interleaved blocks.
Soft-in" refers to the fact that the incoming data may take on values other than 0 or 1, in order to indicate reliability. "Soft-out" refers to the fact that each bit in the decoded output also takes on a value indicating reliability.
Performance of CIRC: [7] CIRC conceals long bust errors by simple linear interpolation. 2.5 mm of track length (4000 bits) is the maximum completely correctable burst length. 7.7 mm track length (12,300 bits) is the maximum burst length that can be interpolated.
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