\end{array}$        \end{array}$        \end{array}$        \end{array}$        \end{array}$        \end{array}$        \end{array}$        \end{array}$        \end{array}$        \end{array}$        \end{array}$        \end{array}$        \end{array}$        \end{array}$
  -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

  : A basis for each one of the fifteen 3-subspaces of the set ${{\cal D}}$ for ${{\mathbb{F}}}_2^4$[]{data-label="tab:3_xubspaces"}

Let $\alpha$ be a primitive element in ${{\mathbb{F}}}_q$. The next structure that has to be considered is a set of $q^2$ different matrices of size $2 \times (q+1)$ over ${{\mathbb{F}}}_q$. These matrices must satisfy the following properties:

1.  Let $v_1,v_2,\ldots , v_{q+1}$ be the $q+1$ consecutive columns of such a matrix. For each $i$, $3 \leq i \leq q+1$, $v_i = \alpha^{i-3} v_1 + v_2$ (a scalar $\beta$ is multiplied by each element of a vector $v$ in the product $\beta v$, and the vector addition $v_1 + v_2$ is performed element by element in ${{\mathbb{F}}}_q$.).

2.  The set of $q^2$ matrices form a linear subspace of dimension two over ${{\mathbb{F}}}_q$.

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

Since these $q^2$ matrices form a linear subspace, it follows that there union is a linear code. In the sequel, this code will be 