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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]
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
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 ...
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 ...
A self-dual code is one which is its own dual. This implies that n is even and dim C = n/2.If a self-dual code is such that each codeword's weight is a multiple of some constant >, then it is of one of the following four types: [1]
The parity bit may be used within another constituent code. In an example using the DVB-S2 rate 2/3 code the encoded block size is 64800 symbols (N=64800) with 43200 data bits (K=43200) and 21600 parity bits (M=21600). Each constituent code (check node) encodes 16 data bits except for the first parity bit which encodes 8 data bits.
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.
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.