# Parity-check matrix

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 satisfy. It can be used to decide if a particular vector is a codeword and is also used in decoding algorithms.

## Definition

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] That is, they show how linear combinations of certain digits (components) of each codeword equal zero. For example, the parity check matrix

$H = \left[ \begin{array}{cccc} 0&0&1&1\\ 1&1&0&0 \end{array} \right]$,

compactly represents the parity check equations,

\begin{align} c_3 + c_4 &= 0 \\ c_1 + c_2 &= 0 \end{align},

that must be satisfied for the vector $(c_1, c_2, c_3, c_4)$ to be a codeword of C.

## Creating a parity check matrix

The parity check matrix for a given code can be derived from its generator matrix (and vice-versa).[3] If the generator matrix for an [n,k]-code is in standard form

$G = \begin{bmatrix} I_k | P \end{bmatrix}$,

then the parity check matrix is given by

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

because

$G H^{\top} = P-P = 0$.

Negation is performed in the finite field Fq. Note that if the characteristic of the underlying field is 2 (i.e., 1 + 1 = 0 in that field), as in binary codes, then -P = P, so the negation is unnecessary.

For example, if a binary code has the generator matrix

$G = \left[ \begin{array}{cc|ccc} 1&0&1&0&1 \\ 0&1&1&1&0 \\ \end{array} \right]$,

then its parity check matrix is

$H = \left[ \begin{array}{cc|ccc} 1&1&1&0&0 \\ 0&1&0&1&0 \\ 1&0&0&0&1 \\ \end{array} \right]$.

## Syndromes

For any (row) vector x of the ambient vector space, s = Hx is called the syndrome of x. The vector x is a codeword if and only if s = 0. The calculation of syndromes is the basis for the syndrome decoding algorithm.[4]