论文标题
关于切割盖子及其代码
On cutting blocking sets and their codes
论文作者
论文摘要
令PG $（R，Q）$为有限字段$ {\ rm GF}（Q）$上的$ r $二维投影空间。 PG $（R，Q）$的点$ \ cal x $是削减阻滞集，如果对于PG $的每个超平面$π$（R，Q）$ The Set $π\ CAP \ CAL x $ spans $π$。切割阻滞集会引起饱和集和最小的线性代码，而尺寸尽可能小的代码特别感兴趣。我们观察到，从Fancsali和Sziklai获得的切割阻滞集，使用一组成对的分离线，就产生了最小的线性代码，其长度相对于其维度线性增长。我们还提供了两个不同的构造：PG $（3，Q^3）$ 3（q+1）（q^2+1）$的切割阻塞集，作为三个成对分离$ q $ q $ ordor subster子地球图的结合，以及pg $ 7（q+1）$ 7（q+1）$ 7（q+1）的切割块，来自desargues $ $ $ $ $（q+1）$（q+1）。在这两种情况下，所获得的切割集合都比已知的切割组小。作为副产品，我们进一步改善了某些饱和集的最小尺寸的上限以及最小$ q $ - Q $ - ARY线性代码的最小长度，其尺寸为$ 4 $和$ 6 $。
Let PG$(r, q)$ be the $r$-dimensional projective space over the finite field ${\rm GF}(q)$. A set $\cal X$ of points of PG$(r, q)$ is a cutting blocking set if for each hyperplane $Π$ of PG$(r, q)$ the set $Π\cap \cal X$ spans $Π$. Cutting blocking sets give rise to saturating sets and minimal linear codes and those having size as small as possible are of particular interest. We observe that from a cutting blocking set obtained by Fancsali and Sziklai, by using a set of pairwise disjoint lines, there arises a minimal linear code whose length grows linearly with respect to its dimension. We also provide two distinct constructions: a cutting blocking set of PG$(3, q^3)$ of size $3(q+1)(q^2+1)$ as a union of three pairwise disjoint $q$-order subgeometries and a cutting blocking set of PG$(5, q)$ of size $7(q+1)$ from seven lines of a Desarguesian line spread of PG$(5, q)$. In both cases the cutting blocking sets obtained are smaller than the known ones. As a byproduct we further improve on the upper bound of the smallest size of certain saturating sets and on the minimum length of a minimal $q$-ary linear code having dimension $4$ and $6$.
