komm.SimplexCode

Simplex (maximum-length) code. For a given parameter $\kappa \geq 2$, it is the linear block code with generator matrix whose columns are all the $2^\kappa - 1$ nonzero binary $\kappa$-tuples. The simplex code (also known as maximum-length code) has the following parameters:

• Length: $n = 2^\kappa - 1$
• Dimension: $k = \kappa$
• Redundancy: $m = 2^\kappa - \kappa - 1$
• Minimum distance: $d = 2^{\kappa - 1}$

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

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

Notes

• For $\kappa = 2$ it reduces to the single parity check code of length $3$.
• Its dual is the Hamming code.
• Simplex codes are constant-weight codes.

Attributes:

• kappa (int) –

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

• extended (bool) –

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

This function constructs the code in systematic form, with the information set on the left.

Examples:

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

>>> code = komm.SimplexCode(3, extended=True)
>>> (code.length, code.dimension, code.redundancy)
(8, 4, 4)
>>> code.minimum_distance
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]])