$V=V(r,q^n)$ such that $k>r$ and $\dim_{{{\mathbb F}}_q}(M\cap U)<k-1$ for every $(r-1)$-dimensional ${\mathbb{F}_{q^n}}$-subspace $M$ of $V$. Then the $k$-dimensional ${{\mathbb F}}_q$-subspace $W+\Gamma^{\perp}$ of the quotient space $\operatorname{\mathbb{V}}/\Gamma^{\perp}$ will be denoted by $ U^{\perp_{D}}$ and will be called the *Delsarte dual* of $U$ (w.r.t. $\perp$).

The Delsarte duality preserves the property of being scattered w.r.t. ${\mathbb{F}_{q^n}}$-subspaces, in the following sense.

[@CsMPZ Theorem 3.3] \[thm:dual\] Let $U$ be a $k$-dimensional $h$-scattered ${{\mathbb F}}_q$-subspace of a vector space $V=V(r,q^n)$ with $n\geq h+3$. Then $U^{\perp_{D}}$ is an $\frac{rn}{h+1}$-dimensional $(n-h-2)$-scattered ${{\mathbb F}_{q}}$-subspace of $\operatorname{\mathbb{V}}/\Gamma^\perp=V(k-r,q^n)$.

Proposition \[prop:property\] points out some properties of the Delsarte duality.

\[prop:property\] Let $U$, $W$, $V$, $\Gamma$, $\operatorname{\mathbb{V}}$, $\perp$ and $\perp_D$ be defined as above. The following properties hold:

-   $(U^{\perp_D})^{\perp_D}=W+\Gamma=\varphi^{-1}(U)$;

-   under the assumption $n\geq h+3$, $U$ is an $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}_{q}}$-subspace of $V$ if and only if $U^{\perp_D}$ is an $\frac{rn}{h+1}$-dimensional $(n-h-2)$-scattered ${{\mathbb F}_{q}}$-subspace of $\operatorname{\mathbb{V}}/\Gamma^{\perp}$.

The first property easily follows from the definition of Delsarte duality. Together with Theorem \[thm:dual\] applied to $U^{\perp_D}$, this yields the second property.

Generalities on codes {#sec:codes}
---------------------

In this section we recall some properties of codes that will be used in the paper. In Section \[sec:Hamming\] we consider ${{\mathbb F}_{q}}$-linear codes with respect 