A note on short minimal codes from subgeometries

TL;DR

This paper provides a concise proof for the existence of certain minimal linear codes derived from subgeometries in projective spaces, focusing on odd prime powers greater than 9.

Contribution

It offers a simplified proof for the existence of minimal codes from subgeometries in PG(3,q^3) for odd q > 9, building on prior geometric results.

Findings

Existence of minimal codes with parameters [3(q^2+1)(q+1),4]_{q^3} for odd q > 9

Short proof leveraging small blocking sets in projective planes

Extension of geometric methods to code theory

Abstract

In a 2022, Bartoli, Cossidente, Marino, and Pavese proved that in the projective space ${\rm PG}(3,q^3)$, 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.