komm.HammingCode

Hamming code. For a given parameter $\mu \geq 2$, it is the linear block code with check matrix whose columns are all the $2^\mu - 1$ nonzero binary $\mu$-tuples. The Hamming code has the following parameters:

• Length: $n = 2^\mu - 1$
• Dimension: $k = 2^\mu - \mu - 1$
• Redundancy: $m = \mu$
• Minimum distance: $d = 3$

In its extended version, the Hamming code has the following parameters:

• Length: $n = 2^\mu$
• Dimension: $k = 2^\mu - \mu - 1$
• Redundancy: $m = \mu + 1$
• Minimum distance: $d = 4$

For more details, see LC04, Sec. 4.1.

Notes

• For $\mu = 2$ it reduces to the repetition code of length $3$.
• Its dual is the simplex code.
• Hamming codes are perfect codes.

Parameters:

• mu (int) –

  The parameter $\mu$ of the code. Must satisfy $\mu \geq 2$.

• extended (bool) –

  Whether to use the extended version of the Hamming code. Default is False.

This class represents the code in systematic form, with the information set on the left.

Examples:

>>> code = komm.HammingCode(3)
>>> (code.length, code.dimension, code.redundancy)
(7, 4, 3)
>>> code.generator_matrix
array([[1, 0, 0, 0, 1, 1, 0],
       [0, 1, 0, 0, 1, 0, 1],
       [0, 0, 1, 0, 0, 1, 1],
       [0, 0, 0, 1, 1, 1, 1]])
>>> code.check_matrix
array([[1, 1, 0, 1, 1, 0, 0],
       [1, 0, 1, 1, 0, 1, 0],
       [0, 1, 1, 1, 0, 0, 1]])
>>> code.minimum_distance()
3

>>> code = komm.HammingCode(3, extended=True)
>>> (code.length, code.dimension, code.redundancy)
(8, 4, 4)
>>> code.generator_matrix
array([[1, 0, 0, 0, 1, 1, 0, 1],
       [0, 1, 0, 0, 1, 0, 1, 1],
       [0, 0, 1, 0, 0, 1, 1, 1],
       [0, 0, 0, 1, 1, 1, 1, 0]])
>>> code.check_matrix
array([[1, 1, 0, 1, 1, 0, 0, 0],
       [1, 0, 1, 1, 0, 1, 0, 0],
       [0, 1, 1, 1, 0, 0, 1, 0],
       [1, 1, 1, 0, 0, 0, 0, 1]])
>>> code.minimum_distance()
4