galois.bch_generator_matrix

class galois.bch_generator_matrix(n, k, c=1, primitive_poly=None, primitive_element=None, systematic=True)

Returns the generator matrix for the primitive binary \(\textrm{BCH}(n, k)\) code.

Parameters

  • n (int) – The codeword size \(n\), must be \(n = 2^m - 1\).

  • k (int) – The message size \(k\).

  • c (int, optional) – The first consecutive power of \(\alpha\). The default is 1.

  • primitive_poly (galois.Poly, optional) – Optionally specify the primitive polynomial that defines the extension field \(\mathrm{GF}(2^m)\). The default is None which uses the lexicographically-smallest primitive polynomial, i.e. galois.primitive_poly(2, m, method="smallest"). The use of the lexicographically-smallest primitive polynomial, as opposed to a Conway polynomial, is most common for the default in textbooks, Matlab, and Octave.

  • primitive_element (int, galois.Poly, optional) – Optionally specify the primitive element \(\alpha\) whose powers are roots of the generator polynomial \(g(x)\). The default is None which uses the lexicographically-smallest primitive element in \(\mathrm{GF}(2^m)\), i.e. galois.primitive_element(2, m).

  • systematic (bool, optional) – Optionally specify if the encoding should be systematic, meaning the codeword is the message with parity appended. The default is True.

Returns

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

Return type

galois.FieldArray

Examples

In [1]: galois.bch_generator_matrix(15, 7)
Out[1]: 
GF([[1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0],
    [0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0],
    [0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0],
    [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1],
    [0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0],
    [0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1],
    [0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1]], order=2)

In [2]: galois.bch_generator_matrix(15, 7, systematic=False)
Out[2]: 
GF([[1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
    [0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0],
    [0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0],
    [0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0],
    [0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0],
    [0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0],
    [0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1]], order=2)