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  2. Generator matrix - Wikipedia

    en.wikipedia.org/wiki/Generator_matrix

    where is the identity matrix and P is a () matrix. When the generator matrix is in standard form, the code C is systematic in its first k coordinate positions. [3] A generator matrix can be used to construct the parity check matrix for a code

  3. Parity-check matrix - Wikipedia

    en.wikipedia.org/wiki/Parity-check_matrix

    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]

  4. Dual code - Wikipedia

    en.wikipedia.org/wiki/Dual_code

    A generator matrix for the dual code is the parity-check matrix for the original code and vice versa. The dual of the dual code is always the original code.

  5. Reed–Solomon error correction - Wikipedia

    en.wikipedia.org/wiki/Reed–Solomon_error...

    This matrix is a Vandermonde matrix over . In other words, the Reed–Solomon code is a linear code , and in the classical encoding procedure, its generator matrix is A {\displaystyle A} . Systematic encoding procedure: The message as an initial sequence of values

  6. Reed–Muller code - Wikipedia

    en.wikipedia.org/wiki/Reed–Muller_code

    The generator matrix [ edit ] 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).

  7. Low-density parity-check code - Wikipedia

    en.wikipedia.org/wiki/Low-density_parity-check_code

    LDPC codes functionally are defined by a sparse parity-check matrix. This sparse matrix is often randomly generated, subject to the sparsity constraints—LDPC code construction is discussed later. These codes were first designed by Robert Gallager in 1960. [5] Below is a graph fragment of an example LDPC code using Forney's factor graph notation.

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  9. Systematic code - Wikipedia

    en.wikipedia.org/wiki/Systematic_code

    For a systematic linear code, the generator matrix, , can always be written as = [|], where is the ... is the identity matrix of size . Examples ...