A strong blocking set in a finite projective space is a set of points that intersects each hyperplane in a spanning set. We provide a new graph theoretic construction of such sets: combining constant-degree expanders with asymptotically good codes, we explicitly construct strong blocking sets in the $(k-1)$-dimensional projective space over $\mathbb{F}_q$ that have size $O( q k )$. Since strong blocking sets have recently been shown to be equivalent to minimal linear codes, our construction gives the first explicit construction of $\mathbb{F}_q$-linear minimal codes of length $n$ and dimension $k$, for every prime power $q$, for which $n = O (q k)$. This solves one of the main open problems on minimal codes.

翻译：在有限射影空间中，强阻挡集是一类能与每个超平面相交并生成该超平面的点集。我们提出了一种基于图论的新构造方法：通过将恒定度扩展图与渐近优码相结合，我们在$\mathbb{F}_q$上的$(k-1)$维射影空间中显式构造了大小为$O( q k )$的强阻挡集。由于近期研究表明强阻挡集与最小线性码等价，我们的构造首次为任意素数幂$q$给出了长度为$n$、维数为$k$的$\mathbb{F}_q$线性最小码的显式构造，其中$n = O (q k)$。这解决了最小码研究中的核心开放问题之一。