Explicit strong blocking sets and minimal codes from expander graphs using a new graph parameter

Source abstract

A strong blocking set is a set of points in a finite projective space that intersects every hyperplane of the space in a spanning set of the hyperplane. These objects have been studied both in finite geometry and coding theory due to their equivalence with minimal codes. We give an explicit construction of strong blocking sets using expander graphs that improves the explicit construction of Alon et al. (Noga Alon, Anurag Bishnoi, Shagnik Das, Alessandro Neri, Strong blocking sets and minimal codes from expander graphs. Transactions of the American Mathematical Society ) by introducing a new geometrically motivated graph parameter that is better suited for this problem. Using this approach, we improve the constant $C$ in the explicit construction of a strong blocking set of size $C (q+1) k$ in the $(k - 1)$-dimensional projective space over $\mathbb{F}_q$ for every prime power $q \geq 7$.