Converts the generator polynomial $g(x)$ into the generator matrix $\mathbf{G}$ for an $[n,k]$ cyclic code.

Parameters¶

n: int¶

The codeword size $n$.

generator_poly: Poly¶

The generator polynomial $g(x)$.

systematic: bool = True¶

Optionally specify if the encoding should be systematic, meaning the codeword is the message with parity appended. The default is True.

Returns¶

The $(k,n)$ generator matrix $\mathbf{G}$, such that given a message $\mathbf{m}$, a codeword is defined by $\mathbf{c}=\mathbf{m}\mathbf{G}$.

Examples¶

Compute the generator matrix for the $\mathrm{Hamming}(7,4)$ code.

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

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

In [3]: galois.poly_to_generator_matrix(7, g, systematic=True)
Out[3]: 
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)