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komm.SystematicBlockCode

Systematic linear block code. A systematic linear block code is a linear block code in which the information bits can be found in predefined positions in the codeword, called the information set $\mathcal{K}$, which is a $k$-sublist of $[0 : n)$; the remaining positions are called the parity set $\mathcal{M}$, which is a $m$-sublist of $[0 : n)$. In this case, the generator matrix then has the property that the columns indexed by $\mathcal{K}$ are equal to $I_k$, and the columns indexed by $\mathcal{M}$ are equal to $P$. The check matrix has the property that the columns indexed by $\mathcal{M}$ are equal to $I_m$, and the columns indexed by $\mathcal{K}$ are equal to $P^\transpose$. The matrix $P \in \mathbb{B}^{k \times m}$ is called the parity submatrix of the code.

The constructor expects the parity submatrix and the information set.

Attributes:

  • parity_submatrix –

    The parity submatrix $P$ the code, which is a $k \times m$ binary matrix.

  • information_set –

    Either an array containing the indices of the information positions, which must be a $k$-sublist of $[0 : n)$, or one of the strings 'left' or 'right'. The default value is 'left'.

Examples:

>>> code = komm.SystematicBlockCode(parity_submatrix=[[0, 1, 1], [1, 1, 0]])
>>> (code.length, code.dimension, code.redundancy)
(5, 2, 3)
>>> code.generator_matrix
array([[1, 0, 0, 1, 1],
       [0, 1, 1, 1, 0]])
>>> code.check_matrix
array([[0, 1, 1, 0, 0],
       [1, 1, 0, 1, 0],
       [1, 0, 0, 0, 1]])

>>> code = komm.SystematicBlockCode(parity_submatrix=[[0, 1, 1], [1, 1, 0]], information_set='right')
>>> (code.length, code.dimension, code.redundancy)
(5, 2, 3)
>>> code.generator_matrix
array([[0, 1, 1, 1, 0],
       [1, 1, 0, 0, 1]])
>>> code.check_matrix
array([[1, 0, 0, 0, 1],
       [0, 1, 0, 1, 1],
       [0, 0, 1, 1, 0]])

length: int property

The length $n$ of the code.

dimension: int property

The dimension $k$ of the code.

redundancy: int property

The redundancy $m$ of the code.

rate: float property

The rate $R = k/n$ of the code.

generator_matrix: npt.NDArray[np.integer] cached property

The generator matrix $G \in \mathbb{B}^{k \times n}$ of the code.

generator_matrix_right_inverse: npt.NDArray[np.integer] cached property

The right-inverse $G^+ \in \mathbb{B}^{n \times k}$ of the generator matrix.

check_matrix: npt.NDArray[np.integer] cached property

The check matrix $H \in \mathbb{B}^{m \times n}$ of the code.

encode

Applies the encoding mapping $\Enc : \mathbb{B}^k \to \mathbb{B}^n$ of the code. This method takes one or more sequences of messages and returns their corresponding codeword sequences.

Parameters:

  • input (ArrayLike) –

    The input sequence(s). Can be either a single sequence whose length is a multiple of $k$, or a multidimensional array where the last dimension is a multiple of $k$.

Returns:

  • output (NDArray[integer]) –

    The output sequence(s). Has the same shape as the input, with the last dimension expanded from $bk$ to $bn$, where $b$ is a positive integer.

inverse_encode

Applies the inverse encoding mapping $\Enc^{-1} : \mathbb{B}^n \to \mathbb{B}^k$ of the code. This method takes one or more sequences of codewords and returns their corresponding message sequences.

Parameters:

  • input (ArrayLike) –

    The input sequence(s). Can be either a single sequence whose length is a multiple of $n$, or a multidimensional array where the last dimension is a multiple of $n$.

Returns:

  • output (NDArray[integer]) –

    The output sequence(s). Has the same shape as the input, with the last dimension contracted from $bn$ to $bk$, where $b$ is a positive integer.

check

Applies the check mapping $\mathrm{Chk}: \mathbb{B}^n \to \mathbb{B}^m$ of the code. This method takes one or more sequences of received words and returns their corresponding syndrome sequences.

Parameters:

  • input (ArrayLike) –

    The input sequence(s). Can be either a single sequence whose length is a multiple of $n$, or a multidimensional array where the last dimension is a multiple of $n$.

Returns:

  • output (NDArray[integer]) –

    The output sequence(s). Has the same shape as the input, with the last dimension contracted from $bn$ to $bm$, where $b$ is a positive integer.

codewords cached

Returns the codewords of the code. This is a $2^k \times n$ matrix whose rows are all the codewords. The codeword in row $i$ corresponds to the message obtained by expressing $i$ in binary with $k$ bits (MSB in the right).

codeword_weight_distribution cached

Returns the codeword weight distribution of the code. This is an array of shape $(n + 1)$ in which element in position $w$ is equal to the number of codewords of Hamming weight $w$, for $w \in [0 : n]$.

minimum_distance cached

Returns the minimum distance $d$ of the code. This is equal to the minimum Hamming weight of the non-zero codewords.

coset_leaders cached

Returns the coset leaders of the code. This is a $2^m \times n$ matrix whose rows are all the coset leaders. The coset leader in row $i$ corresponds to the syndrome obtained by expressing $i$ in binary with $m$ bits (MSB in the right), and whose Hamming weight is minimal. This may be used as a LUT for syndrome-based decoding.

coset_leader_weight_distribution cached

Returns the coset leader weight distribution of the code. This is an array of shape $(n + 1)$ in which element in position $w$ is equal to the number of coset leaders of weight $w$, for $w \in [0 : n]$.

packing_radius cached

Returns the packing radius of the code. This is also called the error-correcting capability of the code, and is equal to $\lfloor (d - 1) / 2 \rfloor$.

covering_radius cached

Returns the covering radius of the code. This is equal to the maximum Hamming weight of the coset leaders.