Finding error patterns from a syndrome I have a parity-check matrix 
$$
H= 
        \left[ \begin{matrix}
        1 & 1 & 1 & 0 & 1 & 0 & 0 & 0\\
        0 & 1 & 1 & 1 & 0 & 1 & 0 & 0\\
        1 & 0 & 1 & 1 & 0 & 0 & 1 & 0\\
        1 & 1 & 0 & 1 & 0 & 0 & 0 & 1\\
        \end{matrix} \right]
$$
and a received codeword $r = (01110110)$ 
I computed the syndrome $s = (0010)$, which tells that the error pattern should be $e = (00000010)$. 
Now my problem is, can I find other error patterns from this syndrome?
To be general, do we get syndromes that isn't a vector of the generator matirx? If yes, then how do we find out the error patterns?
 A: Given that the weight of the error pattern should be minimal, you have the only correct error pattern for this syndrome in this code.
You could find $(01011000)$ as error pattern from this syndrome, but that pattern obviously has more weight, and less chance of being the correct error pattern.
It would also result in a diffent decoded word.
I'm not sure what the minimum distance of the original code was, probably 4, in that case when 2 errors would occur, you would find multiple error patterns for the same syndrome.
In that case we would say the code is 1 error correcting and 2 error detecting. (A second error could be detected but not corrected)
A: If syndrome $s$ corresponds to an error pattern $e$, then the set 
$\mathcal B$ of
all vectors that have the same syndrome $s$ is given by
$$\mathcal B = \{e + c\colon c \in \mathcal C\}$$
where $\mathcal C$ is the set of all codewords.  No
codeword can have a nonzero syndrome (alternatively, if a
vector has syndrome $\mathbf 0$, then that vector is a
codeword in $\mathcal C$). If you compute the
difference between two vectors that have
a particular syndrome, that difference is a codeword.
