Catch & Correct

Syndrome Names the Place

The failed parity checks add up to the changed bit position.

highlighted = computed this step

Recheck the parity groups

The receiver runs the same parity groups on the received word. Failed groups form the syndrome.

failed checks form syndrome\text{failed checks form syndrome}
Syndrome names the placeHamming receive and correction rows recomputed from pinned bits.Syndrome names the place - 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

Three check results

The recomputed syndrome bits are 0, 1, and 1.

s1=0,s2=1,s4=1s_1=0,\quad s_2=1,\quad s_4=1
Syndrome names the placeHamming receive and correction rows recomputed from pinned bits.Syndrome names the place - 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

Add the failed group numbers

The failed checks add to position 6. That number is the bit position to flip back.

2+4=62+4=6
Syndrome names the placeHamming receive and correction rows recomputed from pinned bits.Syndrome names the place - 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

The syndrome is 6, so this toy points at the changed position exactly.

syndrome=6\text{syndrome}=6
Syndrome names the placeHamming receive and correction rows recomputed from pinned bits.Syndrome names the place - 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