$\mathbf{forall}i\in \; \{1,\; ...,m-1\}\mathbf{do}$
$\phantom{nn}\mathbf{G}[0;i(n+1)]\; =\; 1$
$\mathbf{forall}i\in \; \{1,\; ...,m-2\}\mathbf{do}$
$\phantom{nn}\mathbf{G}[1;i(n+1)+1]\; =\; 1$

$\phantom{,}\mathrm{f_{m_s}}\leftarrow \mathrm{f_{m_s}}+{\ell }_{{𝔽}_q^{m^{\times }}}+{\ell }_{{𝔽}_q^{n^{\times }}}$