Linear codes and syndrome


Assume a linear code with (4,2) where we want to encode 2-bit data to 4-bit data. The generator (G) matrix is

1 0 0 0 0 1 1 0 

Now, if we want to encode 00, we get

[0 0] * [1 0 0 0] = [0 0 0 0]         [0 1 1 0] 

Also the parity check matrix (H) matrix is

0 1 1 0 0 0 0 1 

and assume the received data is 0100 where a single bit error occurs on the second bit (from left to right).

Multiplying H.C_received, we get

            [0] [0 1 1 0] * [1]  = [10] [0 0 0 1]   [0]             [0] 

So the syndrome is not zero means there is an error in the received data. BUT, the value of syndrome 10 matches second and third column of the H matrix.

So, how do we find out exactly that the second bit is faulty?