Build the Codeword

Parity Bits Watch Groups

Even parity fills the check positions from overlapping groups.

highlighted = computed this step

Parity slots are separate

The check bits live in positions 1, 2, and 4. They are not extra data.

parity positions 1,2,4\text{parity positions }1,2,4
Parity bits watch groupsHamming encode and correction rows recomputed from pinned bits.Parity bits watch groups - codeword 0110011, syndrome 6rowvaluecheckdata3=1,5=0,6=1,7=1positions 3,5,6,7p10positions 1,3,5,7 evenp21positions 2,3,6,7 evenp40positions 4,5,6,7 evencodeword0110011positions 1..7flip6received bit is toggledreceived0110001one changed bits10positions 1,3,5,7s21positions 2,3,6,7s41positions 4,5,6,7syndrome6s4+s2+s1 as a positioncorrected0110011flip syndrome position

The first parity group

The first parity row watches positions 1, 3, 5, and 7. Even parity makes its check bit 0.

p1=0p_1=0
Parity bits watch groupsHamming encode and correction rows recomputed from pinned bits.Parity bits watch groups - codeword 0110011, syndrome 6rowvaluecheckdata3=1,5=0,6=1,7=1positions 3,5,6,7p10positions 1,3,5,7 evenp21positions 2,3,6,7 evenp40positions 4,5,6,7 evencodeword0110011positions 1..7flip6received bit is toggledreceived0110001one changed bits10positions 1,3,5,7s21positions 2,3,6,7s41positions 4,5,6,7syndrome6s4+s2+s1 as a positioncorrected0110011flip syndrome position

The other checks overlap

The second parity bit becomes 1, and the fourth parity bit becomes 0. The groups overlap on purpose.

p2=1,p4=0p_2=1,\quad p_4=0
Parity bits watch groupsHamming encode and correction rows recomputed from pinned bits.Parity bits watch groups - codeword 0110011, syndrome 6rowvaluecheckdata3=1,5=0,6=1,7=1positions 3,5,6,7p10positions 1,3,5,7 evenp21positions 2,3,6,7 evenp40positions 4,5,6,7 evencodeword0110011positions 1..7flip6received bit is toggledreceived0110001one changed bits10positions 1,3,5,7s21positions 2,3,6,7s41positions 4,5,6,7syndrome6s4+s2+s1 as a positioncorrected0110011flip syndrome position

Summary

Even parity means each watched group has an even count of one bits after its parity bit is filled.

even parity checks each group\text{even parity checks each group}
Parity bits watch groupsHamming encode and correction rows recomputed from pinned bits.Parity bits watch groups - codeword 0110011, syndrome 6rowvaluecheckdata3=1,5=0,6=1,7=1positions 3,5,6,7p10positions 1,3,5,7 evenp21positions 2,3,6,7 evenp40positions 4,5,6,7 evencodeword0110011positions 1..7flip6received bit is toggledreceived0110001one changed bits10positions 1,3,5,7s21positions 2,3,6,7s41positions 4,5,6,7syndrome6s4+s2+s1 as a positioncorrected0110011flip syndrome position