We prove new upper bounds on the smallest size of affine blocking sets, that is, sets of points in a finite affine space that intersect every affine subspace of a fixed codimension. We show an equivalence between affine blocking sets with respect to codimension-$2$ subspaces that are generated by taking a union of lines through the origin, and strong blocking sets in the corresponding projective space, which in turn are equivalent to minimal codes. Using this equivalence, we improve the current best upper bounds on the smallest size of a strong blocking set in finite projective spaces over fields of size at least $3$. Furthermore, using coding theoretic techniques, we improve the current best lower bounds on strong blocking set. Over the finite field $\mathbb{F}_3$, we prove that minimal codes are equivalent to linear trifferent codes. Using this equivalence, we show that any linear trifferent code of length $n$ has size at most $3^{n/4.55}$, improving the recent upper bound of Pohoata and Zakharov. Moreover, we show the existence of linear trifferent codes of length $n$ and size at least $\frac{1}{3}\left( 9/5 \right)^{n/4}$, thus (asymptotically) matching the best lower bound on trifferent codes. We also give explicit constructions of affine blocking sets with respect to codimension-$2$ subspaces that are a constant factor bigger than the best known lower bound. By restricting to $\mathbb{F}_3$, we obtain linear trifferent codes of size at least $3^{7n/240}$, improving the current best explicit construction that has size $3^{n/112}$.

翻译：我们证明了仿射阻塞集最小大小的新上界，即有限仿射空间中与固定余维数仿射子空间均相交的点集。我们揭示了由过原点的直线并集生成的余维数为2的子空间所对应的仿射阻塞集与相应射影空间中的强阻塞集之间的等价性，而后者等价于极小码。借助这一等价性，我们改进了在至少包含3个元素的有限域上射影空间中强阻塞集最小大小的当前最优上界。此外，利用编码理论技术，我们改进了强阻塞集的当前最优下界。在有限域$\mathbb{F}_3$上，我们证明了极小码等价于线性三值区分码。利用这一等价性，我们证明了长度为$n$的线性三值区分码的大小至多为$3^{n/4.55}$，改进了Pohoata和Zakharov的最新上界。进一步地，我们证明了长度为$n$且大小至少为$\frac{1}{3}\left( 9/5 \right)^{n/4}$的线性三值区分码的存在性，从而（渐近地）匹配了三值区分码的最佳下界。我们还给出了余维数为2的子空间所对应的仿射阻塞集的显式构造，其大小比已知最佳下界大一个常数因子。通过限制到$\mathbb{F}_3$，我们得到了大小至少为$3^{7n/240}$的线性三值区分码，改进了当前最佳显式构造（其大小为$3^{n/112}$）。