论文标题
在最小尺寸的线性集上
On linear sets of minimum size
论文作者
论文摘要
An $\mathbb{F}_q$-linear set of rank $k$ on a projective line $\mathrm{PG}(1,q^h)$, containing at least one point of weight one, has size at least $q^{k-1}+1$ (see [J. De Beule and G. Van De Voorde, The minimum size of a linear set, J. Comb. Theory, Ser: A 164 (2019), 109-124。]）。这样的一组的经典例子是由俱乐部给出的。在本文中，我们构建了一个符合此下限的广泛线性套件，我们可以在其中规定最重的点的重量，以介绍$ k/2 $和$ k-1 $之间的任何值。我们的构造扩展了$ q^{k-1}+1 $的线性示例，以$ \ mathrm {pg}（1，q^h）$构建为$ k = h = 4 $ [G。 Bonoli和O. Polverino，$ \ Mathbb {F} _Q $ -Linear Blocking Sets在$ \ Mathrm {pg}（2，Q^4）$，Innov中。发病率吉姆。 2（2005），35--56。]和$ k = h $ in [G.。 Lunardon和O. Polverino。大小$ q^t+q^{t-1}+1 $的阻塞集。 J.梳子。理论，SER：A 90（2000），148-158。]。我们确定构造的线性集的重量分布，并将其描述为子几幅的投影。对于小$ k $，我们调查了所有尺寸$ q^{k-1}+1 $的线性套件是否来自我们的施工。 最后，我们修改构造以定义大小$ q^{k-1}+q^{k-2}+\ ldots+q^{k-l}+1 $ in $ \ mathrm {pg}（l，q）$中的线性。这导致了新的无限封闭套装的新无限家庭，这些家族不是Rédei类型的。
An $\mathbb{F}_q$-linear set of rank $k$ on a projective line $\mathrm{PG}(1,q^h)$, containing at least one point of weight one, has size at least $q^{k-1}+1$ (see [J. De Beule and G. Van De Voorde, The minimum size of a linear set, J. Comb. Theory, Ser: A 164 (2019), 109-124.]). The classical example of such a set is given by a club. In this paper, we construct a broad family of linear sets meeting this lower bound, where we are able to prescribe the weight of the heaviest point to any value between $k/2$ and $k-1$. Our construction extends the known examples of linear sets of size $q^{k-1}+1$ in $\mathrm{PG}(1,q^h)$ constructed for $k=h=4$ [G. Bonoli and O. Polverino, $\mathbb{F}_q$-Linear blocking sets in $\mathrm{PG}(2,q^4)$, Innov. Incidence Geom. 2 (2005), 35--56.] and $k=h$ in [G. Lunardon and O. Polverino. Blocking sets of size $q^t+q^{t-1}+1$. J. Comb. Theory, Ser: A 90 (2000), 148-158.]. We determine the weight distribution of the constructed linear sets and describe them as the projection of a subgeometry. For small $k$, we investigate whether all linear sets of size $q^{k-1}+1$ arise from our construction. Finally, we modify our construction to define linear sets of size $q^{k-1}+q^{k-2}+\ldots+q^{k-l}+1$ in $\mathrm{PG}(l,q)$. This leads to new infinite families of small minimal blocking sets which are not of Rédei type.
