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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...

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

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...

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...

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...

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

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...

{\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...

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 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...

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...

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

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...

{\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...

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...

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...

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)....

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

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...

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...