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Turbo coding is an iterated soft-decoding scheme that combines two or more relatively simple convolutional codes and an interleaver to produce a block code that can perform to within a fraction of a decibel of the Shannon limit.
The on-line textbook: Information Theory, Inference, and Learning Algorithms, by David J.C. MacKay, contains chapters on elementary error-correcting codes; on the theoretical limits of error-correction; and on the latest state-of-the-art error-correcting codes, including low-density parity-check codes, turbo codes, and fountain codes.
Proof. We need to prove that if you add a burst of length to a codeword (i.e. to a polynomial that is divisible by ()), then the result is not going to be a codeword (i.e. the corresponding polynomial is not divisible by ()).
In 1950, Hamming introduced the [7,4] Hamming code. It encodes four data bits into seven bits by adding three parity bits. As explained earlier, it can either detect and correct single-bit errors or it can detect (but not correct) both single and double-bit errors.
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]
This algorithm is critical to modern iteratively-decoded error-correcting codes, including turbo codes and low-density parity-check codes. Steps involved [ edit ]
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 two-dimensional parity-check code, usually called the optimal rectangular code, is the most popular form of multidimensional parity-check code. Assume that the goal is to transmit the four-digit message "1234", using a two-dimensional parity scheme.