galois.generator_to_parity_check_matrix¶

galois.generator_to_parity_check_matrix(G)

Converts the generator matrix $$\mathbf{G}$$ of a linear $$[n, k]$$ code into its parity-check matrix $$\mathbf{H}$$.

The generator and parity-check matrices satisfy the equations $$\mathbf{G}\mathbf{H}^T = \mathbf{0}$$.

Parameters

G (galois.FieldArray) – The $$(k, n)$$ generator matrix $$\mathbf{G}$$ in systematic form $$\mathbf{G} = [\mathbf{I}_{k,k}\ |\ \mathbf{P}_{k,n-k}]$$.

Returns

The $$(n-k, n)$$ parity-check matrix $$\mathbf{H} = [-\mathbf{P}_{k,n-k}^T\ |\ \mathbf{I}_{n-k,n-k}]$$.

Return type

galois.FieldArray

Examples

In [1]: g = galois.primitive_poly(2, 3); g
Out[1]: Poly(x^3 + x + 1, GF(2))

In [2]: G = galois.poly_to_generator_matrix(7, g); G
Out[2]:
GF([[1, 0, 0, 0, 1, 0, 1],
[0, 1, 0, 0, 1, 1, 1],
[0, 0, 1, 0, 1, 1, 0],
[0, 0, 0, 1, 0, 1, 1]], order=2)

In [3]: H = galois.generator_to_parity_check_matrix(G); H
Out[3]:
GF([[1, 1, 1, 0, 1, 0, 0],
[0, 1, 1, 1, 0, 1, 0],
[1, 1, 0, 1, 0, 0, 1]], order=2)

In [4]: G @ H.T
Out[4]:
GF([[0, 0, 0],
[0, 0, 0],
[0, 0, 0],
[0, 0, 0]], order=2)