论文标题
大$（k; r，s; n，q）$ - 设置在投影空间
Large $(k; r, s; n, q)$-sets in Projective Spaces
论文作者
论文摘要
a $（k; r，s; n，q）$ - set（short：$（r，s）$ - set）$ \ mathrm {pg}（n，q）$是一组点$ x $，带有$ | x | = k $，因此没有$ s $ -space包含$ x $的$ r $点。我们研究了$（R，S）$的渐近尺寸 - 固定$ n $的套件和$ Q \ rightarrow \ infty $。特别是，我们表明存在$（3，2）$ - 尺寸$（1+O（1））Q^{3/2} $的$ N = 6 $，$（4，2）$ - 尺寸$（1+O（1+O（1））Q^{\ frac {\ frac {n-1}} {2}}} $（1+（1+）$（1+（9，2 size）Q^Q^{1+O（1） $ n = 4 $。我们还从1947年开始将Rao的限制概括，并表明$（r，s）$ - set最多具有$ o（q^{\ frac {n-e+1}} {e}}} $，如果存在整数$ d，e \ geq 2 $，则$ d（e \ geq 2）
A $(k; r, s; n, q)$-set (short: $(r,s)$-set) of $\mathrm{PG}(n, q)$ is a set of points $X$ with $|X| = k$ such that no $s$-space contains more than $r$ points of $X$. We investigate the asymptotic size of $(r, s)$-sets for $n$ fixed and $q \rightarrow \infty$. In particular, we show the existence of $(3, 2)$-sets of size $(1+o(1)) q^{3/2}$ for $n=6$, $(4, 2)$-sets of size $(1+o(1)) q^{\frac{n-1}{2}}$, and $(9, 2)$-sets of size $(1+o(1)) q^2$ for $n=4$. We also generalize a bound by Rao from 1947 and show that an $(r,s)$-set has size at most $O(q^{\frac{n-e+1}{e}})$ if there exist integers $d,e \geq 2$ such that $s=d(e-1)$ and $r=de-1$.
