komm.ReedMullerCode

Reed–Muller code. It is a linear block code defined by two integers $\rho$ and $\mu$, which must satisfy $0 \leq \rho < \mu$. See references for more details. The resulting code is denoted by $\mathrm{RM}(\rho, \mu)$, and has the following parameters:

• Length: $n = 2^{\mu}$
• Dimension: $k = 1 + {\mu \choose 1} + \cdots + {\mu \choose \rho}$
• Redundancy: $m = 1 + {\mu \choose 1} + \cdots + {\mu \choose \mu - \rho - 1}$
• Minimum distance: $d = 2^{\mu - \rho}$

For more details, see LC04, Sec. 4.3.

Notes

• For $\rho = 0$ it reduces to a repetition code.
• For $\rho = 1$ it reduces to a lengthened simplex code.
• For $\rho = \mu - 2$ it reduces to an extended Hamming code.
• For $\rho = \mu - 1$ it reduces to a single parity check code.

Attributes:

• rho (int) –

  The parameter $\rho$ of the code.

• mu (int) –

  The parameter $\mu$ of the code.

The parameters must satisfy $0 \leq \rho < \mu$.

Examples:

>>> code = komm.ReedMullerCode(1, 5)
>>> (code.length, code.dimension, code.redundancy)
(32, 6, 26)
>>> code.minimum_distance
16
>>> code.generator_matrix
array([[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1],
       [0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1],
       [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1],
       [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
       [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]])

reed_partitions property

The Reed partitions of the code. See LC04, Sec. 4.3.

Examples:

>>> code = komm.ReedMullerCode(2, 4)
>>> code.reed_partitions[1]
array([[ 0,  1,  4,  5],
       [ 2,  3,  6,  7],
       [ 8,  9, 12, 13],
       [10, 11, 14, 15]])
>>> code.reed_partitions[8]
array([[ 0,  4],
       [ 1,  5],
       [ 2,  6],
       [ 3,  7],
       [ 8, 12],
       [ 9, 13],
       [10, 14],
       [11, 15]])