[8,4,4] Ext. Hamming Code: Realization 1

The [8,8,4] extended Hamming code with codewords

x=[u0, u1, u2, u3, x4, x5, x6, x7]

has a systematic generator matrix G:

|   1 0 0 0   1 1 1 0   |

G = |   0 1 0 0   1 1 0 1   |

|   0 0 1 0   1 0 1 1   |

|   0 0 0 1   0 1 1 1   |

The parity check matrix can easily be derived from G:

|   1 1 1 0   1 0 0 0   |

H = |   1 1 0 1   0 1 0 0   |

|   1 0 1 1   0 0 1 0   |

|   0 1 1 1   0 0 0 1   |

This corresponds to the following four parity checks:

x4 = u0 + u1 + u2
x5 = u0 + u1 + u3
x6 = u0 + u2 + u3
x7 = u1 + u2 + u3.

This yields the following implementation (the function nodes c0 to c7 represent the a priori probabilities of the received bits):

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A more symmetric arrangement of the same realization: