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 whether a particular vector is a codeword and is also used in decoding algorithms.
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.)
[2] That is, they show how linear combinations of certain digits (components) of each codeword equal zero.
to be a codeword of C. From the definition of the parity-check matrix it directly follows the minimum distance of the code is the minimum number d such that every d - 1 columns of a parity-check matrix H are linearly independent while there exist d columns of H that are linearly dependent.
[3] If the generator matrix for an [n,k]-code is in standard form then the parity check matrix is given by because Negation is performed in the finite field Fq.