Gaussian elimination to equations with errors
$q$ is the number of samples and $S$ is the set of indices such that $\sum_{i\in S}x_i = (1,0,...,0)$. In your example, we have $q=3$ and $S=\{1,3\}$. The first bit of the secret $s$ is equals to $\sum_{i\in S}b_i \mod 2$, but with probability $1/2+2^{-\Theta(n)}$ not $1$, because the noises are also considered in this summation. So we need to repeat this procedure $2^{\Theta(n)}$ times to reduce the error probability.