Minimal additive codes and additive strong blocking sets

Abstract

Additive codes over $\mathbb{F}_{q^h}$ generalize linear codes by relaxing linearity over the alphabet while retaining linearity over the subfield $\mathbb{F}_q$. In this paper, we introduce minimal additive codes and we initiate their study from a geometric perspective. We define the concept of additive strong blocking sets, a class of $h$-projective systems whose union forms a strong blocking set. We establish a one-to-one correspondence between equivalence classes of nondegenerate minimal additive codes and equivalence classes of additive strong blocking sets. We also compare this framework with the theory of outer strong blocking sets, showing that the latter arises as a special case. Finally, we provide constructions and existence results for minimal additive codes, and derive upper, lower, and asymptotic bounds on their minimum length.