F}}_q$-subspaces $U^{\perp'}$, where $\perp'$ is defined by the restriction to ${{\mathbb F}}_q$ of any non-degenerate reflexive sesquilinear form over $V$, as defined at the beginning of this section.

### Delsarte duality {#sec:Delsarteduality}

Let $U$ be a $k$-dimensional ${{\mathbb F}}_q$-subspace of a vector space $V=V(r,q^n)$, with $k>r$. By [@LuPo2004 Theorems 1, 2] (see also [@LuPoPo2002 Theorem 1]), there is an embedding of $V$ in $\operatorname{\mathbb{V}}=V(k,q^n)$ with $\operatorname{\mathbb{V}}=V \oplus \Gamma$ for some $(k-r)$-dimensional ${{\mathbb F}}_{q^n}$-subspace $\Gamma$ such that $U={\langle}W,\Gamma{\rangle}_{{{\mathbb F}}_{q}}\cap V$, where $W$ is a $k$-dimensional ${{\mathbb F}}_q$-subspace of $\operatorname{\mathbb{V}}$ satisfying $\langle W\rangle_{{{\mathbb F}}_{q^n}}=\operatorname{\mathbb{V}}$ and $W\cap \Gamma=\{{\bf 0}\}$. Then $ \varphi:V\to\operatorname{\mathbb{V}}/\Gamma$, $\mathbf{v}\mapsto \mathbf{v}+\Gamma$, is an ${\mathbb{F}_{q^n}}$-isomorphism such that $\varphi(U)=W+\Gamma$.

Following [@CsMPZ Section 3], let $\beta'\colon W\times W\rightarrow{{\mathbb F}}_{q}$ be a non-degenerate reflexive sesquilinear form on $W$. Then $\beta'$ can be extended to a non-degenerate reflexive sesquilinear form $\beta\colon \operatorname{\mathbb{V}}\times\operatorname{\mathbb{V}}\rightarrow{{\mathbb F}}_{q^n}$. Let $\perp$ and $\perp'$ be the orthogonal complement maps defined by $\beta$ and $\beta'$ on the lattices of ${{\mathbb F}}_{q^n}$-subspaces of $\operatorname{\mathbb{V}}$ and of ${{\mathbb F}}_q$-subspaces of $W$, respectively. For an ${{\mathbb F}}_q$-subspace $S$ of $W$ the ${{\mathbb F}}_{q^n}$-subspace ${\langle}S {\rangle}_{{{\mathbb F}}_{q^n}}$ of $\operatorname{\mathbb{V}}$ will be denoted by $S^*$. In this case, $(S^*)^{\perp}=(S^{\perp'})^*$.

\[deffff\] Let $U$ be a $k$-dimensional ${{\mathbb F}}_q$-subspace of 