A note on short minimal codes from subgeometries

In a 2022, Bartoli, Cossidente, Marino, and Pavese proved that in the projective space $\mathrm{P}\mathrm{G}(3,q^3)$, one can find three ${\mathbb{F}}_q$-subgeometries such that the union of their point sets is a strong blocking set. This proves the existence of linear minimal codes with parameters $[3(q^2+1)(q+1),4{]}_{q^3}$ for every prime power $q$. We give a short proof of this result for odd values of $q>9$, using the theory of small blocking sets in projective planes.