An additive code of length $n$ over the finite field $F_{q^h}$ is a subset $\mathcal{C}$ of $F_{q^h}^n$ that is closed under addition. Such an additive code is linear over the subfield $F_q$ and therefore has size $q^r$ for some $r$. To denote an additive code we will use the notation $[n, \frac{r}{h}, d]_q^h$, where $d$ is the minimum Hamming distance of $\mathcal{C}$. Additive codes are particularly interesting because, in certain cases, they achieve parameters for which no linear code exists. However, still very few cases are known and the explicit construction of such codes is a challenging problem. Moreover, they play a central role in quantum quantum error correction.

In this talk, we adopt a geometric perspective on additive codes. We introduce the notions of minimal additive codes and additive strong blocking sets in projective spaces. We establish a one-to-one correspondence between these two objects and investigate the resulting structural properties.

Collaboration avec avec Gianira N. Alfarano