A Simple Explicit Construction of Smaller Strong Blocking Sets

Source abstract

Alon, Bishnoi, Das and Neri constructed sequences of strong blocking sets in $(k-1)$-dimension projective space $P({\bf F}_q, k-1)$ with sizes at most $Cqk$, where $k$ goes to the infinity and $C$ is a fixed positive constant not depending on $q$, in their paper published in Transactions AMS, vol. 277, pp. 5389-5410, 2024. This is equivalent to a sequence of minimal $[n_i,k_i]_q$ codes with positive rate $\frac{k_i}{n_i} \geq c$, where $k_i$ goes to the infinity and $c$ is a fixed positive constant only depending on $q$. They solved one of the main open problems on minimal codes. The Alon-Bishnoi-Das-Neri construction was based on asymptotically good sequences of algebraic geometric codes and expander graphs. However, the constant $C$ is large for small prime power $q$. In this paper, based on a simple approach using $(q-1)$-power intersecting algebraic geometric codes via concatenation to construct minimal codes, we construct explicit strong blocking sets in $(k-1)$-dimension projective space with sizes around $2(q^2-1)k$, equivalently, explicit minimal $[n,k]_q$ codes with length around $2(q^2-1)k$. This is an essential improvement on the previous result of Alon-Bishnoi-Das-Neri, when $q \leq 11$. For large $q$, strong blocking sets and minimal codes in our construction are worse than the previous construction of Alon-Bishnoi-Das-Neri. However, our approach is much simpler and does not depend on expander graphs.