are $q^4$ matrices in the extension space. Moreover, these $q^4$ matrices form a linear subspace of dimension four over ${{\mathbb{F}}}_q$. Since the extension code is a linear subspace of dimension two of the extension space, it follows that we can partition the $q^4$ matrices of the extension space into $q^2$ sets of size $q^2$ having the following properties:

1.  The extension code is the first set.

2.  Let $v_1,v_2,\ldots , v_{q+1}$ be the $q+1$ consecutive columns of any matrix in any of the codes. For each $i$, $3 \leq i \leq q+1$, $v_i = \alpha^{i-3} v_1 + v_2$.

3.  For each $i$, $1 \leq i \leq q+1$, the $q^2$ $i$-th column vectors in the $q^2$ matrices, of any of the $q^2$ sets, are all distinct, i.e. they consist of all possible column vectors of length 2.

An example for an extension space (the extension code and its coset) is given in Table \[tab:extension\].

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   $C$   $\begin{array}{c}   $\begin{array}{c}   $\begin{array}{c}   $\begin{array}{c}   $C_1$   $\begin{array}{c}   $\begin{array}{c}   $\begin{array}{c}   $\begin{array}{c}   $C_2$   $\begin{array}{c}   $\begin{array}{c}   $\begin{array}{c}   $\begin{array}{c}   $C_3$   $\begin{array}{c}   $\begin{array}{c}   $\begin{array}{c}   $\begin{array}{c}
              000 \\              011 \\              101 \\              110 \\                      000 \\              011 \\              101 \\              110 \\                      110 \\              101 \\              011 \\              000 \\                      110 \\              