galois.generator_to_parity_check_matrix

galois.generator_to_parity_check_matrix(G: FieldArray) → FieldArray

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

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

Parameters¶

G: FieldArray¶: The $(k, n)$ generator matrix $G$ in systematic form $G = [I_{k, k} | P_{k, n - k}]$ .

Returns¶

The $(n - k, n)$ parity-check matrix $H = [- P_{k, n - k}^{T} | I_{n - k, n - k}]$ .

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)