Catch & Correct

Correct and Honesty

The syndrome correction restores the toy codeword and states its boundary.

highlighted = computed this step

Flip the named position back

Because the syndrome is 6, the correction flips position 6.

correct position 6\text{correct position }6
Correct and state the boundaryHamming receive and correction rows recomputed from pinned bits.Correct and state the boundary - 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 codeword is restored

After that one correction, the word is 0110011, matching the original codeword.

corrected=0110011\text{corrected}=0110011
Correct and state the boundaryHamming receive and correction rows recomputed from pinned bits.Correct and state the boundary - 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

What this toy leaves out

NOTE: toy one-bit correction only. This is not Reed-Solomon, not a burst or noise model, and not production ECC.

toy one-bit correction only\text{toy one-bit correction only}
Correct and state the boundaryHamming receive and correction rows recomputed from pinned bits.Correct and state the boundary - 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

Hamming correction here is exact: encode 0110011, receive 0110001, syndrome 6, corrected 0110011.

0110001→01100110110001\to0110011
Correct and state the boundaryHamming receive and correction rows recomputed from pinned bits.Correct and state the boundary - 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