$$P_i\tau^h=\sum_{j=0}^{k-1}\lambda_{ij}
P\tau^{j+h}=P\tau^h \cdot P_i(\tau),\quad \forall h=0,\dots,N-1$$ and, for $i=1,\dots,m$, this implies that $$\begin{aligned}
H_i&=[(P_i)_t,(P_i\tau)_t,\dots,(P_i\tau^{N-1})_t]\\
&=[(P\cdot P_i(\tau))_t, (P\tau \cdot P_i(\tau))_t,\dots,(P\tau^{N-1} \cdot P_i(\tau))_t]\\
&=P_i(\tau)_t \cdot [P_t, (P\tau)_t ,\dots,(P\tau^{N-1})_t ] \\
&= P_i(\tau)_t\cdot H_1\end{aligned}$$ Let $H_1^\ast$ be the generator matrix of $\mathscr{C}^\perp$ with $\mathscr{C}=<g(x)>$ written as $$H_1^\ast:= \begin{pmatrix} h^\ast (x)\\ xh^\ast (x)
\\ \vdots \\ x^{k-1}h^\ast(x)
\end{pmatrix} =[ J \mid
\widehat{H}_1^\ast ]$$ with $det(J)\not= 0$, where $h^\ast(x)=\theta^{N-k}(h_0)^{-1}x^{N-k}h(x^{-1})$ with $x^N-\alpha=g(x)h(x)$. By hypothesis, $H_1$ is also a parity check matrix of $\mathscr{C}$ which can be written as $H_1=[I_k\mid \widehat{H}_1]$, where $I_k$ is the identity matrix of dimension $k$. This implies that $J^{-1}H_1^\ast=H_1$ and $J^{-1}\widehat{H}_1^\ast =\widehat{H}_1$.\
Let $A$ be a matrix with $N$ columns and define the linear operator $\diamond$ as follows: $$x^{-1}\diamond A := A\cdot \left( \begin{array}{ c  c c | c }
0 &  \cdots &0 &  \alpha \\  \hline
1 &  \cdots &0 & 0 \\
\vdots & \ddots & \vdots & \vdots \\
0 &\cdots & 1  & 0 \\
 \end{array}\right).$$ Observe that $x^{-1}\diamond (A'\cdot B')=A'\cdot
(x^{-1}\diamond B')$ for any two matrices $A'$ and $B'$ such that $A'\cdot B'$ is well defined and $B'$ has $N$ columns.\
Furthermore, for $h=0,\dots,N-1$, we get $$\begin{aligned}
x^{-1}\diamond H_1 &= x^{-1}\diamond [P_t,(P\tau)_t,\dots,(P\tau^{N-1})_t]\\ &
= [(P\tau)_t,\dots,(P\tau^{N-1})_t, \alpha 