H = matrix.įor any (row) vector x of the ambient vector space, s = H x ⊤ is called the syndrome of x. That is, they show how linear combinations of certain digits (components) of each codeword equal zero. The rows of a parity check matrix are the coefficients of the parity check equations. This means that a codeword c is in C if and only if the matrix-vector product H c ⊤ = 0 (some authors would write this in an equivalent form, c H ⊤ = 0.) It can be used to decide whether a particular vector is a codeword and is also used in decoding algorithms.įormally, a parity check matrix H of a linear code C is a generator matrix of the dual code, C ⊥. 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.
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