pcm2gm

sionna.phy.fec.utils.pcm2gm(pcm: numpy.ndarray, verify_results: bool = True) → numpy.ndarray

Generates the generator matrix for a given parity-check matrix.

This function converts the parity-check matrix pcm (denoted as \(\mathbf{H}\)) to systematic form and uses the following relationship to compute the generator matrix \(\mathbf{G}\) over GF(2):

\[\mathbf{G} = [\mathbf{I} | \mathbf{M}] \Rightarrow \mathbf{H} = [\mathbf{M}^T | \mathbf{I}]. \tag{1}\]

This derivation is based on the requirement for an all-zero syndrome:

\[\mathbf{H} \mathbf{c}^T = \mathbf{H} * (\mathbf{u} * \mathbf{G})^T = \mathbf{H} * \mathbf{G}^T * \mathbf{u}^T = \mathbf{0},\]

where \(\mathbf{c}\) represents an arbitrary codeword and \(\mathbf{u}\) the corresponding information bits.

This leads to:

\[\mathbf{G} * \mathbf{H}^T = \mathbf{0}. \tag{2}\]

It can be shown that (1) satisfies (2), as in GF(2):

\[[\mathbf{I} | \mathbf{M}] * [\mathbf{M}^T | \mathbf{I}]^T = \mathbf{M} + \mathbf{M} = \mathbf{0}.\]

Parameters:

• pcm (numpy.ndarray) – Binary parity-check matrix of shape [n - k, n].

• verify_results (bool) – If True, verifies that the generated generator matrix is orthogonal to the parity-check matrix in GF(2).

Outputs:

gm – Binary generator matrix of shape [k, n].

Notes

This function requires pcm to have full rank. An error is raised if pcm does not meet this requirement.