to the Hamming metric in ${{\mathbb F}}_{q}^N$, while in Section \[sec:rank\] we consider ${{\mathbb F}_{q}}$-linear codes with respect to the rank metric in ${{\mathbb F}}_{q}^{m\times n}$.

### Projective systems and linear codes {#sec:Hamming}

Let $\mathcal{C}\subseteq{{\mathbb F}}_q^N$ be an ${{\mathbb F}_{q}}$-linear code of length $N$, dimension $k$ and minimum distance $d$ over the alphabet ${{\mathbb F}_{q}}$; we denote by $[N,k,d]_q$ the parameters of $\mathcal{C}$. A generator matrix of $\mathcal{C}$ is a matrix $G\in{{\mathbb F}}_q^{k\times N}$ whose rows form a basis of $\mathcal{C}$. The weight of a codeword $\mathbf{c}\in\mathcal{C}$ is the number of nonzero components of $\mathbf{c}$, and $A_i^H$ will denote the number of codewords of weight $i$ in $\mathcal{C}$. The $N$-tuple $(A_0^H=1,A_1^H,\ldots,A_N^H)$ is called the weight distribution of $\mathcal{C}$, and the polynomial $\sum_{i=0}^{N}A_i^H z^i$ is the weight enumerator of $\mathcal{C}$.

A projective $[N,k,d]_q$-system is a point subset $\mathcal{P}$ of $\Omega={\mathrm{PG}}(k-1,q)$ of size $N$, not contained in any hyperplane of $\Omega$, such that $$d= N - \max\{|\mathcal{P}\cap\mathcal{H}|\colon \mathcal{H}\,\mbox{ is a hyperlane of }\,\Omega\}.$$ The matrix $G\in{{\mathbb F}}_q^{k\times N}$ whose columns are the coordinates of the points of a projective $[N,k,d]_q$-system $\mathcal{P}$ is the generator matrix of a linear code with parameters $[N,k,d]_q$. Different choices of the coordinates yield linear codes which are 