X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/16dcc.git/blobdiff_plain/36e14b57e13803de510ec32c84df163f86dca6c3..c9d6a79d174af327059607a3e0a1f23ab7921a49:/generating.tex

diff --git a/generating.tex b/generating.tex
index d512a98..0837fda 100644
--- a/generating.tex
+++ b/generating.tex
@@ -51,7 +51,7 @@ The question which remains to solve is:
 The answer is indeed positive. We furthermore have the following strongest 
 result.
 \begin{thrm}
-There exist $b \in \Nats$ such that $\Gamma_{\{b\}}(f)$ is complete.
+There exists $b \in \Nats$ such that $\Gamma_{\{b\}}(f)$ is complete.
 \end{thrm}
 \begin{proof}
 There is an arc $(x,y)$ in the 
@@ -65,13 +65,12 @@ This section ends with the idea of removing a Hamiltonian cycle in the
 $\mathsf{N}$-cube. 
 In such a context, the Hamiltonian cycle is equivalent to a Gray code.
 Many approaches have been proposed a way to build such codes, for instance 
-the Reflected Binary Code. In this one, one of the bits is switched 
-exactly $2^{\mathsf{N}-}$ \ANNOT{formule incomplète : $2^{\mathsf{N}-1}$ ??} for a $\mathsf{N}$-length cycle. 
-
-%%%%%%%%%%%%%%%%%%%%%
-
-The function that is built 
-from the \ANNOT{Phrase non terminée}
+the Reflected Binary Code. In this one and 
+for a $\mathsf{N}$-length cycle, one of the bits is exactly switched 
+$2^{\mathsf{N}-1}$ times whereas the others bits are modified at most 
+$\left\lfloor \dfrac{2^{\mathsf{N-1}}}{\mathsf{N}-1} \right\rfloor$ times.
+It is clear that the function that is built from such a code would
+not provide a uniform output.  
 
 The next section presents how to build balanced Hamiltonian cycles in the 
 $\mathsf{N}$-cube with the objective to embed them into the