an integer $b$ that corresponds to an iteration number (\textit{i.e.}, the length of the walk), and
an initial configuration $x^0$.
Starting from $x^0$, the algorithm repeats $b$ times
-a random choice of which edge to follow, and traverses this edge
+a random choice of which edge to follow, and crosses this edge
provided it is allowed to do so, \textit{i.e.},
when $\textit{Random}(1)$ is not null.
The final configuration is thus outputted.
Notice that the chaos property of $G_f$ given in Sect.\ref{sec:proofOfChaos}
-only requires that the graph $\Gamma_{\{b\}}(f)$ is strongly connected.
+only requires the graph $\Gamma_{\{b\}}(f)$ to be strongly connected.
Since the $\chi_{\textit{16HamG}}$ algorithm
only adds probability constraints on existing edges,
it preserves this property.
has the smallest practical mixing time.
Such functions are
given in Table~\ref{table:nc}.
-In this table, let us consider for instance
+In this table, let us consider, for instance,
the function $\textcircled{a}$ from $\Bool^4$ to $\Bool^4$
defined by the following images :
$[13, 10, 9, 14, 3, 11, 1, 12, 15, 4, 7, 5, 2, 6, 0, 8]$.
In this table the column that is labeled with $b$ %(respectively by $E[\tau]$)
gives the practical mixing time
-where the deviation to the standard distribution is lesser than $10^{-6}$.
+where the deviation to the standard distribution is inferior than $10^{-6}$.
%(resp. the theoretical upper bound of stopping time as described in Sect.~\ref{sec:hypercube}).
\end{tabular}
\end{scriptsize}
\end{center}
-\caption{Functions with DSCC Matrix and smallest MT\label{table:nc}}
+\caption{Functions with DSCC Matrix and smallest MT}\label{table:nc}
\end{table*}
