Catch & Correct

One Bit Flips

The receive side starts with one pinned bit changed.

highlighted = computed this step

The sender has a clean codeword

The clean codeword is 0110011 before the receiver sees it.

sent=0110011\text{sent}=0110011
One bit flipsHamming receive and correction rows recomputed from pinned bits.One bit flips - 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

A pinned bit changes

This toy flips exactly position 6. That changes the received word to 0110001.

flip position 6\text{flip position }6
One bit flipsHamming receive and correction rows recomputed from pinned bits.One bit flips - 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 receiver does not know the clean word

The table shows both words so we can learn the method. In a real receive step, the receiver only has the received word and recomputes checks.

receiver recomputes checks\text{receiver recomputes checks}
One bit flipsHamming receive and correction rows recomputed from pinned bits.One bit flips - 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 received word 0110001 differs by one pinned position in this toy.

received=0110001\text{received}=0110001
One bit flipsHamming receive and correction rows recomputed from pinned bits.One bit flips - 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