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A multidimensional parity-check code (MDPC) is a type of error-correcting code that generalizes two-dimensional parity checks to higher dimensions. It was developed as an extension of simple parity check methods used in magnetic recording systems and radiation-hardened memory designs .
Formally, a parity check matrix H of a linear code C is a generator matrix of the dual code, C ⊥. This means that a codeword c is in C if and only if the matrix-vector product Hc ⊤ = 0 (some authors [1] would write this in an equivalent form, cH ⊤ = 0.) The rows of a parity check matrix are the coefficients of the parity check equations. [2]
Type II codes are binary self-dual codes which are doubly even. Type III codes are ternary self-dual codes. Every codeword in a Type III code has Hamming weight divisible by 3. Type IV codes are self-dual codes over F 4. These are again even. Codes of types I, II, III, or IV exist only if the length n is a multiple of 2, 8, 4, or 2 respectively.
Long code; Low-density parity-check code, also known as Gallager code, as the archetype for sparse graph codes; LT code, which is a near-optimal rateless erasure correcting code (Fountain code) m of n codes; Nordstrom-Robinson code, used in Geometry and Group Theory [31] Online code, a near-optimal rateless erasure correcting code; Polar code ...
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.
The first 4680 data bits are repeated 13 times (used in 13 parity codes), while the remaining data bits are used in 3 parity codes (irregular LDPC code). For comparison, classic turbo codes typically use two constituent codes configured in parallel, each of which encodes the entire input block (K) of data bits.
An example EXIT chart showing two components "right" and "left" and an example decoding (blue) An extrinsic information transfer chart, commonly called an EXIT chart, is a technique to aid the construction of good iteratively-decoded error-correcting codes (in particular low-density parity-check (LDPC) codes and Turbo codes).
A generator matrix for a linear [,,]-code has format , where n is the length of a codeword, k is the number of information bits (the dimension of C as a vector subspace), d is the minimum distance of the code, and q is size of the finite field, that is, the number of symbols in the alphabet (thus, q = 2 indicates a binary code, etc.).