[Math] Finding error patterns from a syndrome

Question

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?

Answer 1

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.