Parity-check matrix

In coding theory, a parity-check matrix of a linear block code C is a generator matrix of the dual code. As such, a codeword c is in C if and only if the matrix-vector product H^Tc=0.

The rows of a parity check matrix are parity checks on the codewords of a code. That is, they show how linear combinations of certain digits of each codeword equal zero. For example, the parity check matrix

$H={\begin{bmatrix}0011\\1100\end{bmatrix}}$

specifies that for each codeword, digits 1 and 2 should sum to zero and digits 3 and 4 should sum to zero.

For more information see Hamming code and generator matrix.

Creating a parity check matrix

The parity check matrix for a given code can be derived from its generator matrix (and vice-versa). If the generator matrix for an [n,k]-code in standard form is

$G={\begin{bmatrix}I_{k}|P\end{bmatrix}}$

the parity check matrix can be calculated as

$H={\begin{bmatrix}-P^{T}|I_{n-k}\end{bmatrix}}$

Negation is performed in the finite field mod $q$. Note that this means in binary codes negation is unnecessary as -1 = 1 (mod 2).

For example, if a binary code has the generator matrix

$G={\begin{bmatrix}10|101\\01|110\\\end{bmatrix}}$

The parity check matrix becomes

$H={\begin{bmatrix}11|100\\01|010\\10|001\\\end{bmatrix}}$

For any valid codeword $x$, $Hx=0$. For any invalid codeword ${\tilde {x}}$, the syndrome $S$ satisfies $H{\tilde {x}}=S$.