Extended linear code

Extended linear code Error control capability of an linear code can be improved by adding an overall parity-check bit. An extended code is obtained. The addition of such bit increases a code minimum distance by one. For the Hamming extended code: , what enables single-error correction and also double error detection. A code parity-check equations are supplemented with an overall parity-check sum:
Then, the parity-check matrix of an extended code can be obtained by adding at the right side, the all-0 column (original code parity-check equations remain unchanged) and then, by adding at the bottom, the all-1 row (overall parity-check).
Decoder calculates two syndromes: original (Hamming) code syndrome and parity-check syndrome:
They designate the decoder decision.
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