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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.
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Low-density parity-check (LDPC) codes are a class of highly efficient linear block codes made from many single parity check (SPC) codes. They can provide performance very close to the channel capacity (the theoretical maximum) using an iterated soft-decision decoding approach, at linear time complexity in terms of their block length.
The rate of a block code is defined as the ratio between its message length and its block length: = /. A large rate means that the amount of actual message per transmitted block is high.
A special case of constant weight codes are the one-of-N codes, that encode bits in a code-word of bits.The one-of-two code uses the code words 01 and 10 to encode the bits '0' and '1'.
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. With the addition of an overall parity bit, it becomes the [8,4] extended Hamming code and can both detect and correct single-bit errors and detect (but not correct) double-bit errors.
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 ()).
Parvaresh–Vardy codes are a family of error-correcting codes first described in 2005 by Farzad Parvaresh and Alexander Vardy. [1] They can be used for efficient list-decoding . See also