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komm.GolayCode

Binary Golay code. It is the linear block code with parity submatrix $$ P = \begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 \end{bmatrix} $$

The Golay code has the following parameters:

  • Length: $23$
  • Dimension: $12$
  • Minimum distance: $7$
Notes
  • The binary Golay code is a perfect code.

Attributes:

  • extended (bool) –

    If True, constructs the code in extended version. The default value is False.

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

Examples:

>>> code = komm.GolayCode()
>>> (code.length, code.dimension, code.redundancy)
(23, 12, 11)
>>> code.minimum_distance()
7

>>> code = komm.GolayCode(extended=True)
>>> (code.length, code.dimension, code.redundancy)
(24, 12, 12)
>>> code.minimum_distance()
8

minimum_distance cached

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