Minimal additive codes and additive strong blocking sets

2026-06-23

AI Summary

This work investigates the construction of nondegenerate minimal additive codes and their intrinsic connection to strong blocking sets from a geometric perspective. By introducing the notion of additive strong blocking sets, the authors establish a one-to-one correspondence between such sets and minimal additive codes, and further unify the theory of exterior strong blocking sets within the framework of $h$-projective systems. Combining additive coding theory over finite fields with methods from projective geometry, the study provides explicit constructions and existence results for these codes, derives upper and lower bounds—as well as asymptotic bounds—on their minimal length, and achieves, for the first time, a systematic integration of minimal additive codes with blocking set theory.

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.

Problem

• minimal additive codes
• additive strong blocking sets
• projective systems
• strong blocking sets
• nondegenerate codes

Innovation

• minimal additive codes
• additive strong blocking sets
• projective systems
• finite fields
• coding theory