Consider the code with following parity-check matrix $$ \left(\begin{array}{ccc}0&1&0&0\\1&0&0&1\end{array} \right) $$ Given the syndrome $s = 11$, what is the coset leader?
I was given this problem on a multiple choice test. Options included $1100$ and $0101$, but multiplication with check matrix results $11$ in both cases, so they both have the same syndrome. Since their weight is the same, how do we choose the coset leader? Correct answer was $1100$, but I'm not sure I understand why.