Hamming's Original Code

Let be defined by

So
So is a Binary Code.

Suppose we receive . We form the syndrome
where

If then
If for some then the place where and differ is given by (not mod 2).
This is since if then the syndrome of is the same as the syndrome of and the syndrome of is the binary expansion of 3 (for example - can check the others as well)

Thus is 1-Error correcting (it is not 2)