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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]
The matrix product of the generator and parity-check matrices, [|] [|], is the matrix of all zeroes, and by intent. Indeed, this is an example of the very definition of any parity check matrix with respect to its 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
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 ...
For example, p 2 provides an even parity for bits 2, 3, 6, and 7. It also details which transmitted bit is covered by which parity bit by reading the column. For example, d 1 is covered by p 1 and p 2 but not p 3 This table will have a striking resemblance to the parity-check matrix (H) in the next section.
The parity-check matrix of a Hamming code is constructed by listing all columns of length r that are non-zero, which means that the dual code of the Hamming code is the shortened Hadamard code, also known as a Simplex code. The parity-check matrix has the property that any two columns are pairwise linearly independent.
For general , the generator matrix of the augmented Hadamard code is a parity-check matrix for the extended Hamming code of length and dimension , which makes the augmented Hadamard code the dual code of the extended Hamming code. Hence an alternative way to define the Hadamard code is in terms of its parity-check matrix: the parity-check ...
The two-dimensional parity-check code, usually called the optimal rectangular code, is the most popular form of multidimensional parity-check code. Assume that the goal is to transmit the four-digit message "1234", using a two-dimensional parity scheme. First the digits of the message are arranged in a rectangular pattern: 12 34