\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 numerof 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} \begin{exampleblock}{Study of previous code} \begin{minipage}{0.48\textwidth} \begin{itemize} \item Let $L^*=000,100,101,001,011,111,110,010$ \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} \begin{minipage}{0.48\textwidth} \includegraphics[scale=0.5]{iter_f0e} \end{minipage} \end{exampleblock}