How to verify that a matrix is a parity check matrix for g?
Answers
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
{\displaystyle H=\left[{\begin{array}{cccc}0&0&1&1\\1&1&0&0\end{array}}\right]},compactly represents the parity check equations,
{\displaystyle {\begin{aligned}c_{3}+c_{4}&=0\\c_{1}+c_{2}&=0\end{aligned}}},that must be satisfied for the vector {\displaystyle (c_{1},c_{2},c_{3},c_{4})} 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 Hare linearly independent while there exist d columns of H that are linearly dependent.