${{\cal B}}$ contains $q$ spreads, and the set ${{\cal C}}$ contains $q^2$ spreads. Any partition of the $q^2+q+1$ spreads is appropriate for this purpose. Such a partition for ${{\mathbb{F}}}_2^4$ is given in Table \[tab:partition\].

In the construction, we have another set ${{\cal D}}$ which contains all the $q^3 +q^2 +q+1$ distinct 3-subspaces of ${{\mathbb{F}}}_q^4$. An example for a basis of the fifteen 3-subspaces of ${{\mathbb{F}}}_2^4$ is given in Table \[tab:3\_xubspaces\].

  -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
   ${{\cal A}}$    $\begin{array}{c}   $\begin{array}{c}   $\begin{array}{c}   $\begin{array}{c}   $\begin{array}{c}                                                                                                                                                                                                                                                                                          
                        000 \\              011 \\              011 \\              011 \\              011 \\                                                                                                                                                                                                                                                                                                
                        011 \\              011 \\              101 \\              101 \\              000 \\                                                                                                                                                                                                                                                                                                
                        011 \\              101 \\              011 \\              000 \\              101 \\                                                                                                                                                                                                                                                                                                
                          101                 000                 011                 101                 110                                                                                                                                                                                                                                                                                                 
                     \end{array}$        \end{array}$        \end{array}$        \end{array}$        \end{array}$                                                                                                                                                                                                                                                                                             
  --------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- -------------------
   $ {{\cal B}}$   $\begin{array}{c}   $\begin{array}{c}   $\begin{array}{c}   $\begin{array}{c}   $\begin{array}{c}   $\begin{array}{c}   $\begin{array}{c}   $\begin{array}{c}   $\begin{array}{c}   $\begin{array}{c}                                                                                                                                                                                      
                        000 \\              011 \\              011 \\              011 \\              011 \\              011 \\              000 \\              011 \\              011 \\              011 \\                                                                                                                                                                                            
                        011 \\              000 \\              101 \\              101 \\              011 \\              101 \\              011 \\              101 \\              011 \\              000 \\                                                                                                                                                                                            
                        101 \\              101 \\              011 \\              101 \\              