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 a field 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 strong blocking sets 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 with size at least $\frac{1}{3}\left( 9/5 \right)^{n/4}$, thus (asymptotically) matching the best lower bound on non-linear 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:在最小大小的松叶封隔套件上,我们证明新的上限, 也就是说, 在固定的封隔套件中, 在固定的封隔区中, 将固定的封隔区中的每个松叶子子空间相互交叉, 也就是说, 在固定的封隔套件中, 我们展示了一个新的上限。 在固定的封隔套件中, 通过线条组合通过源代码生成的, 而在相应的投影空间中, 坚固的封隔套件在最小尺寸上, 这反过来相当于最小的代码 。 使用这个等值, 我们改进了目前最硬的封堵区中, 硬的最小面积在3美元。 此外, 我们改进了Pohoata和Zakharov的最近上限, 最硬的上限在3美元 ; 因此, 我们证明坚固的封屏蔽套件相当于直线的三角码。 使用这个等值, 任何直线三角底的代码在最硬的长度值中, 最多是 3 ⁇ /4.55美元,, 改进了最近最硬的硬的硬的硬的硬的封框框, 。