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komm.BlockCode

General binary linear block code. It is characterized by its generator matrix $G \in \mathbb{B}^{k \times n}$, and by its check matrix $H \in \mathbb{B}^{m \times n}$, which are related by $G H^\transpose = 0$. The parameters $n$, $k$, and $m$ are called the code length, dimension, and redundancy, respectively, and are related by $k + m = n$. For more details, see LC04, Ch. 3.

The constructor expects either the generator matrix or the check matrix.

Parameters:

  • generator_matrix (Array2D[int]) –

    The generator matrix $G$ of the code, which is a $k \times n$ binary matrix.

  • check_matrix (Array2D[int]) –

    The check matrix $H$ of the code, which is a $m \times n$ binary matrix.

Examples:

>>> code = komm.BlockCode(generator_matrix=[[1, 0, 0, 0, 1, 1], [0, 1, 0, 1, 0, 1], [0, 0, 1, 1, 1, 0]])
>>> (code.length, code.dimension, code.redundancy)
(6, 3, 3)
>>> code.generator_matrix
array([[1, 0, 0, 0, 1, 1],
       [0, 1, 0, 1, 0, 1],
       [0, 0, 1, 1, 1, 0]])
>>> code.check_matrix
array([[0, 1, 1, 1, 0, 0],
       [1, 0, 1, 0, 1, 0],
       [1, 1, 0, 0, 0, 1]])

>>> code = komm.BlockCode(check_matrix=[[0, 1, 1, 1, 0, 0], [1, 0, 1, 0, 1, 0], [1, 1, 0, 0, 0, 1]])
>>> (code.length, code.dimension, code.redundancy)
(6, 3, 3)
>>> code.generator_matrix
array([[1, 0, 0, 0, 1, 1],
       [0, 1, 0, 1, 0, 1],
       [0, 0, 1, 1, 1, 0]])
>>> code.check_matrix
array([[0, 1, 1, 1, 0, 0],
       [1, 0, 1, 0, 1, 0],
       [1, 1, 0, 0, 0, 1]])

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.

Examples:

>>> code = komm.BlockCode(generator_matrix=[[1, 0, 0, 0, 1, 1], [0, 1, 0, 1, 0, 1], [0, 0, 1, 1, 1, 0]])
>>> code.rate
0.5

enc_mapping

The encoding mapping $\Enc : \mathbb{B}^k \to \mathbb{B}^n$ of the code. This is a function that takes a message $u \in \mathbb{B}^k$ and returns the corresponding codeword $v \in \mathbb{B}^n$.

Examples:

>>> code = komm.BlockCode(generator_matrix=[[1, 0, 0, 0, 1, 1], [0, 1, 0, 1, 0, 1], [0, 0, 1, 1, 1, 0]])
>>> enc_mapping = code.enc_mapping
>>> enc_mapping([1, 0, 1])
array([1, 0, 1, 1, 0, 1])

inv_enc_mapping

The inverse encoding mapping $\Enc^{-1} : \mathbb{B}^n \to \mathbb{B}^k$ of the code. This is a function that takes a codeword $v \in \mathbb{B}^n$ and returns the corresponding message $u \in \mathbb{B}^k$.

Examples:

>>> code = komm.BlockCode(generator_matrix=[[1, 0, 0, 0, 1, 1], [0, 1, 0, 1, 0, 1], [0, 0, 1, 1, 1, 0]])
>>> inv_enc_mapping = code.inv_enc_mapping
>>> inv_enc_mapping([1, 0, 1, 1, 0, 1])
array([1, 0, 1])

codewords: np.ndarray cached property

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 whose binary representation (MSB in the right) is $i$.

Examples:

>>> code = komm.BlockCode(generator_matrix=[[1, 0, 0, 0, 1, 1], [0, 1, 0, 1, 0, 1], [0, 0, 1, 1, 1, 0]])
>>> code.codewords
array([[0, 0, 0, 0, 0, 0],
       [1, 0, 0, 0, 1, 1],
       [0, 1, 0, 1, 0, 1],
       [1, 1, 0, 1, 1, 0],
       [0, 0, 1, 1, 1, 0],
       [1, 0, 1, 1, 0, 1],
       [0, 1, 1, 0, 1, 1],
       [1, 1, 1, 0, 0, 0]])

codeword_weight_distribution: np.ndarray cached property

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 $w \in [0 : n]$.

Examples:

>>> code = komm.BlockCode(generator_matrix=[[1, 0, 0, 0, 1, 1], [0, 1, 0, 1, 0, 1], [0, 0, 1, 1, 1, 0]])
>>> code.codeword_weight_distribution
array([1, 0, 0, 4, 3, 0, 0])

chk_mapping

The check mapping $\mathrm{Chk}: \mathbb{B}^n \to \mathbb{B}^m$ of the code. This is a function that takes a received word $r \in \mathbb{B}^n$ and returns the corresponding syndrome $s \in \mathbb{B}^m$.

Examples:

>>> code = komm.BlockCode(generator_matrix=[[1, 0, 0, 0, 1, 1], [0, 1, 0, 1, 0, 1], [0, 0, 1, 1, 1, 0]])
>>> chk_mapping = code.chk_mapping
>>> chk_mapping([1, 0, 1, 1, 0, 1])
array([0, 0, 0])

coset_leaders: np.ndarray cached property

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 whose binary representation (MSB in the right) is $i$, and whose Hamming weight is minimal. This may be used as a LUT for syndrome-based decoding.

Examples:

>>> code = komm.BlockCode(generator_matrix=[[1, 0, 0, 0, 1, 1], [0, 1, 0, 1, 0, 1], [0, 0, 1, 1, 1, 0]])
>>> code.coset_leaders
array([[0, 0, 0, 0, 0, 0],
       [0, 0, 0, 1, 0, 0],
       [0, 0, 0, 0, 1, 0],
       [0, 0, 1, 0, 0, 0],
       [0, 0, 0, 0, 0, 1],
       [0, 1, 0, 0, 0, 0],
       [1, 0, 0, 0, 0, 0],
       [1, 0, 0, 1, 0, 0]])

coset_leader_weight_distribution: np.ndarray cached property

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 $w \in [0 : n]$.

Examples:

>>> code = komm.BlockCode(generator_matrix=[[1, 0, 0, 0, 1, 1], [0, 1, 0, 1, 0, 1], [0, 0, 1, 1, 1, 0]])
>>> code.coset_leader_weight_distribution
array([1, 6, 1, 0, 0, 0, 0])

minimum_distance: int cached property

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

Examples:

>>> code = komm.BlockCode(generator_matrix=[[1, 0, 0, 0, 1, 1], [0, 1, 0, 1, 0, 1], [0, 0, 1, 1, 1, 0]])
>>> code.minimum_distance
3

packing_radius: int cached property

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

Examples:

>>> code = komm.BlockCode(generator_matrix=[[1, 0, 0, 0, 1, 1], [0, 1, 0, 1, 0, 1], [0, 0, 1, 1, 1, 0]])
>>> code.packing_radius
1

covering_radius: int cached property

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

Examples:

>>> code = komm.BlockCode(generator_matrix=[[1, 0, 0, 0, 1, 1], [0, 1, 0, 1, 0, 1], [0, 0, 1, 1, 1, 0]])
>>> code.covering_radius
2