\begin{itemize}
\item Lower bound\footnote{T. Feder and C. Subi.
\newblock Nearly tight bounds on the number of hamiltonian circuits of the
  hypercube and generalizations.
\newblock {\em Inf. Process. Lett.}, 109(5):267--272, February 2009.} of number of  Gray codes in $\Bool^n$: $\left(\frac{n*\log2}{e \log \log n}*(1 - o(1))\right)^{2^n}$ (more than $10^{13}$ when n is 6).

\item Restriction to balanced codes:  the number of edges that modify
  the bit $i$ in $\Gamma(f)$  have to be close to each other
\end{itemize}

\vspace{-1em}
\begin{exampleblock}{Study of previous code}
\vspace{-2em}
\begin{minipage}{0.70\textwidth}
\vspace{-4em}
\begin{itemize}
\item $L^*=000,100,101,001,011,111,110,010$
\item Its transition sequence: $S=3,1,3,2,3,1,3,2$ 
%\item Its transition count function: $\textit{TC}_3(1)= \textit{TC}_3(2)=2$ and  $\textit{TC}_3(3)=4$. 
\end{itemize}
\end{minipage}
\vspace{-4em}
\begin{minipage}{0.29\textwidth}
\vspace{-1.7em}
\hspace{-2.8cm}\includegraphics[scale=0.5]{iter_f0e}
\end{minipage}
\end{exampleblock}
