Low-density parity-check (LDPC) codes are a class of error correction codes which (together with the closely related turbo codes) have gained prominence...

In coding theory, a parity-check matrix of a linear block code C is a matrix which describes the linear relations that the components of a codeword must...

word to any other code word is three) and block length 2r − 1. The parity-check matrix of a Hamming code is constructed by listing all columns of length...

information science used for error detection in quantum qubits. A parity measurement checks the equality of two qubits to return a true or false answer, which...

positions. A generator matrix can be used to construct the parity check matrix for a code (and vice versa). If the generator matrix G is in standard form...

parity-check matrix, being composed of a Vandermonde matrix V {\displaystyle V} and diagonal matrix D {\displaystyle D} , shares the form with check matrices...

two Hamming matrices can be defined: the code generator matrix G and the parity-check matrix H: G T := ( 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1 1 0 1 0 0 0 0 1...

the subcode nodes denote rows of the parity-check matrix H. The digit nodes represent the columns of the matrix H. An edge connects a subcode node to...

square matrix or related classes of matrix Routh–Hurwitz matrix, a square matrix constructed with coefficients of a real polynomial Parity-check matrix is...

in 1986 by Harald Niederreiter. It applies the same idea to the parity check matrix, H, of a linear code. Niederreiter is equivalent to McEliece from...

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...

rank of the parity check matrix is n − k {\displaystyle n-k} . Another simple proof follows from observing that the rows of any generator matrix in standard...

errors. Its check matrix in standard form is [ H 0 0 H ] {\displaystyle {\begin{bmatrix}H&0\\0&H\end{bmatrix}}} where H is the parity-check matrix of the Hamming...

{\displaystyle \dim C+\dim C^{\perp }=n.} A generator matrix for the dual code is the parity-check matrix for the original code and vice versa. The dual of...

methods, including dual check data computations (parity and Reed–Solomon), orthogonal dual parity check data and diagonal parity, have been used to implement...

In physics, a parity transformation (also called parity inversion) is the flip in the sign of one spatial coordinate. In three dimensions, it can also...

A multidimensional parity-check code (MDPC) is a type of error-correcting code that generalizes two-dimensional parity checks to higher dimensions. It...

{\displaystyle n} and minimum distance d {\displaystyle d} with parity-check matrix H {\displaystyle H} . Then clearly C {\displaystyle C} is capable...

{\displaystyle [n,n-m]_{2}\,} linear block code whose parity check matrix is the adjacency matrix of a bipartite expander graph. These codes have good...

Hadamard code is in terms of its parity-check matrix: the parity-check matrix of the Hadamard code is equal to the generator matrix of the Hamming code. Hadamard...

recipient of this medal. The reverse side of the medal depicts a Hamming parity check matrix for a Hamming error-correcting code. Hamming, Richard W. (1962)....

needed] The ternary Golay code consists of 36 = 729 codewords. Its parity check matrix is [ 2 2 2 1 1 0 1 0 0 0 0 2 2 1 2 0 1 0 1 0 0 0 2 1 2 0 2 1 0 0...

n-dimensional code C in an m-dimensional binary vector space, a parity check matrix is an (m − n) × m matrix H having the property that xHT = 0 if and only if x is...

a systematic generator matrix whose non-identity part takes k = 32771 bits. The private key, a quasi-cyclic parity-check matrix with d = 274 nonzero entries...

is performed using a [2048,1723]2 low-density parity-check code on 1723 bits, with the parity check matrix construction based on a generalized Reed–Solomon...

original Srivastava code over GF(q) of length n is defined by a parity check matrix H of alternant form [ α 1 μ α 1 − w 1 ⋯ α n μ α n − w 1 ⋮ ⋱ ⋮ α 1...

marking. Matrix 2 of 5 can encode only digits 0-9. Matrix 2 of 5 can include optional check digit. Most of barcode readers support this symbology. Matrix 2 of...

{\displaystyle g_{1},\ldots ,g_{n-k}} function in the same way as a parity check matrix does for a classical linear block code. One of the fundamental notions...

impractical algorithm becomes practical. See, for example, Low-density parity-check codes, below. An impractical algorithm can still demonstrate that conjectured...

width between 1 and 4 areas. Parity for the digits from left and right group: OEOOEE EEEEEE (O = Odd parity, E = Even parity). The first digit in the EAN...