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:在最小尺寸的埃芬封隔装置上,我们证明有新的上限, 也就是说, 在固定的封隔装置中, 在固定的封隔空间中, 在固定的封隔空间中, 在固定的封隔空间中, 将固定的折叠的每片面积相互交叉。 我们展示了在固定的封隔装置上, 与通过源代码结合产生的折叠- $2$ 的封隔装置和相应的投影空间中的坚硬封隔装置之间的等值。 使用这个等值, 我们改进了目前最小型的封隔空间中, 在固定的封隔隙空间中, 将硬性封隔断的每片面积比 固定的至少3美元 。 此外, 我们展示了最小的最小代码 $7\\\\\\\\\\\\\\\\\ 直立方的线性三角码 。 使用这个等值, 任何长长度的线性三角码, 最多是 $3 ⁇ /4.55}, 改进了最近已知的波霍塔和Zakhar 美元的上限的上限, 美元。