All minimal $[9,4]_{2}$-codes are hyperbolic quadrics

TL;DR

This paper proves that all minimal codes with parameters [9,4]_2 are hyperbolic quadrics by showing that all strong blocking sets of size 9 in PG(3,2) are hyperbolic quadrics, confirming a specific geometric characterization.

Contribution

It establishes that all minimal [9,4]_2-codes correspond to hyperbolic quadrics, providing a complete geometric classification in this case.

Findings

All strong blocking sets of size 9 in PG(3,2) are hyperbolic quadrics.

Confirms the geometric nature of minimal codes with these parameters.

Supports the link between minimal codes and geometric structures in projective spaces.

Abstract

Minimal codes are being intensively studied in last years. $[n,k]_{q}$-minimal linear codes are in bijection with strong blocking sets of size $n$ in $PG(k-1,q)$ and a lower bound for the size of strong blocking sets is given by $(k-1)(q+1)\leq n$. In this note we show that all strong blocking sets of length 9 in $PG(3,2)$ are the hyperbolic quadrics $Q^{+}(3,2)$.