P_t]\\&
=[(P\tau)_t,\dots,(P\tau^{N-1})_t,  (P\tau^N)_t]\\&=(\tau)_t\cdot H_1\\
x^{-2}\diamond H_1&=x^{-1}\diamond (x^{-1}\diamond H_1) \\
&=x^{-1}((\tau)_t \cdot H_1)\\
&=(\tau^2)_t\cdot H_1\\
&\ \ \vdots\\
x^{-h}\diamond  H_1&=(\tau^h)_t\cdot H_1\\\end{aligned}$$

Hence, we have $$\begin{aligned}
H_i&= P_i(\tau)_t \cdot H_1=
P_i(x^{-1})\diamond (J^{-1}\cdot H_1^\ast)
=J^{-1}\cdot (P_i(x^{-1})\diamond H_1^\ast)=\\
&=J^{-1}\cdot \left(P_i(x^{-1})\diamond  \begin{pmatrix}
h^\ast (x)\\ xh^\ast (x) \\ \vdots \\ x^{k-1}h^\ast(x)
\end{pmatrix}\right)=J^{-1}\cdot  \begin{pmatrix}
P_i(x^{-1})\cdot h^\ast (x)\\ P_i(x^{-1})\cdot xh^\ast (x) \\ \vdots \\ P_i(x^{-1})\cdot x^{k-1}h^\ast(x)
\\
\end{pmatrix} \\
& =J^{-1}\cdot \begin{pmatrix}
P_i(x^{-1})\cdot h^\ast (x)\\ x\cdot P_i(x^{-1})h^\ast (x) \\
\vdots \\ x^{k-1}\cdot P_i(x^{-1})h^\ast(x)\end{pmatrix}
=:J^{-1}\cdot
H^\ast_i \\\end{aligned}$$

Therefore, we conclude that $$H=[H_1|H_2|\cdots |H_m]=[J^{-1}H_1^\ast|J^{-1}H_2^\ast|\cdots |J^{-1}H_m^\ast]
=J^{-1}[H_1^\ast|H_2^\ast|\cdots |H_m^\ast],$$ where $\begin{pmatrix}
P_i(x^{-1})\cdot h^\ast (x)\\ x\cdot P_i(x^{-1})h^\ast (x) \\
\vdots \\ x^{k-1}\cdot P_i(x^{-1})h^\ast(x)\end{pmatrix}$, i.e. $[H_1|H_2|\cdots |H_m]$ and $[H_1^\ast|H_2^\ast|\cdots |H_m^\ast]$ are two generator matrices of the same $[mN,k]_q$-code.

The following MAGMA Program $5$ defines the function +SQT(m,N,g)+ which gives an application of the above result :

\[Prog3\]  

    SQT:=function(m,N,g)
    g1:=R!g; d:=Degree(g1); if d ge 1 then
    TT:=R![0,1]; CM:=VerticalJoin(HorizontalJoin(ZeroMatrix(F,
    d-1,1), ScalarMatrix(F,d-1,1)), Matrix(F,1,d,[-Eltseq(g1)[k]: 
    k in {1..d}])); PS<[x]>:=ProjectiveSpace(F,d-1);
    PointsPS:={PS!p : p in Points(Scheme(PS,[0]))}; 
    PointsPS2:=PointsPS; OrbN:={}; repeat
    S:=Random(PointsPS2); PP:=Matrix(F,1,d,[S[k]: k in {1..d}]); 
    jj:=0; repeat
    jj:=jj+1; PP:=Matrix(F,1,d,[SpecialEvaluate(TT,PP[1][i]):
    i in {1..d}])*CM; PPP:=PS![PP[1][i]: i in {1..d}]; 
    PointsPS2:=PointsPS2 diff {PPP}; until S eq PPP; if jj eq N then
    OrbN:=OrbN join {S}; end if;
    until PointsPS2 eq {}; P1:=HorizontalJoin(Matrix(F,1,1,[1]),
    ZeroMatrix(F,1,d-1)); V1:=PS![P1[1][i]: i in 