called the *extension code*.

For each power of a prime $q$ there exists an extension code.

We start with two $2 \times (q+1)$ matrices over ${{\mathbb{F}}}_q$ which will be the basis of the code.

For the first matrix $M_1$ the first column will be ${\tiny \left( \hspace{-0.1cm} \begin{array}{c} 1 \\ 0 \end{array} \hspace{-0.1cm} \right)}$ and the second column will be ${\tiny \left( \hspace{-0.1cm} \begin{array}{c} 0 \\ 1 \end{array} \hspace{-0.1cm} \right)}$. The $i$-th column, $3 \leq i \leq q+1$, is $\alpha^{i-3}
{\tiny \left( \hspace{-0.1cm} \begin{array}{c} 1 \\ 0 \end{array} \hspace{-0.1cm} \right)}+
{\tiny \left( \hspace{-0.1cm} \begin{array}{c} 0 \\ 1 \end{array} \hspace{-0.1cm} \right)}=
{\tiny \left( \hspace{-0.1cm} \begin{array}{c} \alpha^{i-3} \\ 1 \end{array} \hspace{-0.1cm} \right)}$.

For the second matrix $M_2$ the first column will be ${\tiny \left( \hspace{-0.1cm} \begin{array}{c} 0 \\ 1 \end{array} \hspace{-0.1cm} \right)}$ and the second column will be ${\tiny \left( \hspace{-0.1cm} \begin{array}{c} 1 \\ \beta \end{array} \hspace{-0.1cm} \right)}$, where $\beta \in {{\mathbb{F}}}_q$. The $i-$th column, $3 \leq i \leq q+1$, is $\alpha^{i-3}
{\tiny \left( \hspace{-0.1cm} \begin{array}{c} 0 \\ 1 \end{array} \hspace{-0.1cm} \right)}+
{\tiny \left( \hspace{-0.1cm} \begin{array}{c} 1 \\ \beta \end{array} \hspace{-0.1cm} \right)}=
{\tiny \left( \hspace{-0.1cm} \begin{array}{c} 1 \\ \alpha^{i-3} + \beta \end{array} \hspace{-0.1cm} \right)}$. We have to prove that there exists a 