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  2. 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]

  3. Expander code - Wikipedia

    en.wikipedia.org/wiki/Expander_code

    In coding theory, an expander code is a [,] linear block code whose parity check matrix is the adjacency matrix of a bipartite expander graph.These codes have good relative distance (), where and are properties of the expander graph as defined later, rate (), and decodability (algorithms of running time () exist).

  4. Linear code - Wikipedia

    en.wikipedia.org/wiki/Linear_code

    A matrix H representing a linear function : whose kernel is C is called a check matrix of C (or sometimes a parity check matrix). Equivalently, H is a matrix whose null space is C . If C is a code with a generating matrix G in standard form, G = [ I k ∣ P ] {\displaystyle {\boldsymbol {G}}=[I_{k}\mid P]} , then H = [ − P T ∣ I n − k ...

  5. Binary Goppa code - Wikipedia

    en.wikipedia.org/wiki/Binary_Goppa_code

    For practical purposes, parity-check matrix of a binary Goppa code is usually converted to a more computer-friendly binary form by a trace construction, that converts the -by-matrix over () to a -by-binary matrix by writing polynomial coefficients of () elements on successive rows.

  6. Multidimensional parity-check code - Wikipedia

    en.wikipedia.org/wiki/Multidimensional_parity...

    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 .

  7. Ternary Golay code - Wikipedia

    en.wikipedia.org/wiki/Ternary_Golay_code

    The ternary Golay code consists of 3 6 = 729 codewords. Its parity check matrix is [].Any two different codewords differ in at least 5 positions. Every ternary word of length 11 has a Hamming distance of at most 2 from exactly one codeword.

  8. Singleton bound - Wikipedia

    en.wikipedia.org/wiki/Singleton_bound

    In the linear code case a different proof of the Singleton bound can be obtained by observing that rank of the parity check matrix is . [4] Another simple proof follows from observing that the rows of any generator matrix in standard form have weight at most n − k + 1 {\displaystyle n-k+1} .

  9. Reed–Muller code - Wikipedia

    en.wikipedia.org/wiki/Reed–Muller_code

    RM(m − 1, m) codes are single parity check codes of length N = 2 m, rate = and minimum distance =. RM( m − 2, m ) codes are the family of extended Hamming codes of length N = 2 m with minimum distance d min = 4 {\displaystyle d_{\min }=4} .