To make this article self-contained, this section recalls
-basis of the Syndrome Treillis Codes (STC).
+the basis of the Syndrome Treillis Codes (STC).
Let
$x=(x_1,\ldots,x_n)$ be the $n$-bits cover vector of the image $X$,
$m$ be the message to embed, and
$m$ for a given binary matrix $H$.
Let us explain this embedding on a small illustrative example where
-$\rho_X(i,x,y)$ is identically equal to 1,
+$\rho_X(i,x,y)$ is equal to 1,
whereas $m$ and $x$ are respectively a 3 bits column
vector and a 7 bits column vector.
Let then $H$ be the binary Hamming matrix
the solving algorithm has a linear complexity with respect to $n$.
First of all, Filler \emph{et al.} compute the matrix $H$
-by placing a small sub-matrix $\hat{H}$ of size $h × w$ next
-to each other and shifted down by one row.
+by placing a small sub-matrix $\hat{H}$ next
+to each other and by shifting down by one row.
Thanks to this special form of $H$, one can represent
-every solution of $m=Hy$ as a path through a trellis.
+any solution of $m=Hy$ as a path through a trellis.
-Next, the process of finding $y$ consists of two stages: a forward and a backward part.
+Next, the process of finding $y$ consists in two stages: a forward and a backward part.
\begin{enumerate}
\item Forward construction of the trellis that depends on $\hat{H}$, on $x$, on $m$, and on $\rho$.
\item Backward determination of $y$ that minimizes $D$, starting with
