$t$-Fold $s$-Blocking Sets and $s$-Minimal Codes

Blocking sets and minimal codes have been studied for many years in projective geometry and coding theory. In this paper, we provide a new lower bound on the size of $t$-fold $s$-blocking sets without the condition $t \leq q$, which is stronger than the classical result of Beutelspacher in 1983. Then a lower bound on lengths of projective $s$-minimal codes is also obtained. It is proved that $(s+1)$-minimal codes are certainly $s$-minimal codes. We generalize the Ashikhmin-Barg condition for minimal codes to $s$-minimal codes. Many infinite families of $s$-minimal codes satisfying and violating this generalized Ashikhmin-Barg condition are constructed. We also give several examples which are binary minimal codes, but not $2$-minimal codes.

💡 Research Summary

This paper establishes a profound connection between two central concepts in discrete mathematics: t-fold s-blocking sets in projective geometry and s-minimal codes in coding theory. The authors systematically generalize the well-known theory of minimal codes (corresponding to s=1) to higher-dimensional subcodes.

The primary contribution is a new lower bound on the size of a t-fold s-blocking set in PG(k-1, q) (Theorem 3.1). This bound, |B| ≥ t(q^k - 1)/(q^{k-s} - 1), is stronger than the classical 1983 result by Beutelspacher, as it removes the restrictive condition t ≤ q. The proof employs a double counting argument combined with an application of the Griesmer bound for generalized Hamming weights of the linear code generated by the complement of the blocking set. Further refined bounds are given based on whether the complement set spans the entire space.

A key theoretical achievement is the geometric characterization of s-minimal codes (Theorem 4.2). The paper proves that a projective