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. Our main motivation for these new bounds is their application to trifferent codes, which are sets of ternary codes of length $n$ with the property that for any three distinct codewords there is a coordinate where they all have distinct values. 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 $3^{23n/312}$, 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度的每个仿射子空间相交的点集。我们证明了由通过原点的直线并集生成的余维-2子空间仿射阻塞集与对应射影空间中的强阻塞集等价，而后者又等价于最小码。利用这一等价性，我们改进了域大小至少为3的有限射影空间中强阻塞集最小大小的当前最佳上界。此外，使用编码理论技术，我们改进了强阻塞集的当前最佳下界。这些新界的主要动机是它们在三叉码中的应用，即长度为n的三元码集合，满足对于任意三个不同码字，存在一个坐标使得它们取值互异。在有限域$\mathbb{F}_3$上，我们证明最小码等价于线性三叉码。利用该等价性，我们证明任何长度为n的线性三叉码大小至多为$3^{n/4.55}$，改进了Pohoata和Zakharov的最新上界。此外，我们证明了存在长度为n且大小至少为$\frac{1}{3}\left( 9/5 \right)^{n/4}$的线性三叉码，从而（渐近地）匹配了三叉码的最佳下界。我们还给出了关于余维-2子空间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。通过限制在$\mathbb{F}_3$上，我们得到了大小至少为$3^{23n/312}$的线性三叉码，改进了当前大小为$3^{n/112}$的最佳显式构造。