  The $l$-th stopping redundancy $\rho_l(\mathcal C)$ of the binary $[n, k, d]$
code $\mathcal C$, $1 \le l \le d$, is defined as the minimum number of rows in
the parity-check matrix of $\mathcal C$, such that the smallest stopping set is
of size at least $l$. The stopping redundancy $\rho(\mathcal C)$ is defined as
$\rho_d(\mathcal C)$. In this work, we improve on the probabilistic analysis of
stopping redundancy, proposed by Han, Siegel and Vardy, which yields the best
bounds known today. In our approach, we judiciously select the first few rows
in the parity-check matrix, and then continue with the probabilistic method. By
using similar techniques, we improve also on the best known bounds on
$\rho_l(\mathcal C)$, for $1 \le l \le d$. Our approach is compared to the
existing methods by numerical computations.
