空间$\mathrm{PG}(3,q)$和$\mathrm{PG}(4,q)$中已知最小完全帽尺寸的表、界和图形
Tables, bounds and graphics of the smallest known sizes of complete caps in the spaces $\mathrm{PG}(3,q)$ and $\mathrm{PG}(4,q)$
摘要：
本文给出并分析了$q$阶有限域上的维数为$N=3$和$N=4$的射影空间$\mathrm{PG}(N,q)$中小完备帽的计算结果.结果已经得到使用随机贪婪算法和算法与固定顺序的点（FOP）。计算是在$q$值的相对宽的区域中进行的;在$N=3,4$的文献中没有考虑这样宽的区域。新的完整的瓶盖是已知的最小的。在此基础上，我们得到了$t_2(N,q)$的新的上界，即$\mathrm{PG}(N,q)$，特别是\begin{align*} &t_{2}(N,q)<\sqrt{N+2}\cdot q^{\frac{N-1}{2^{\vphantom{H}}}}\sqrt{\ln q},\quad q\in L_{N},\quad N=3,4,\\ &t_{2}(N,q)<\left(\sqrt{N+1}+\frac{1.3}{\ln (2q)}\right)q^{\frac{N-1}{2^{\vphantom{H}}}}\sqrt{\ln q},\quad q\in L_{N},\quad N=3,4, \end{align*}，其中\begin{align*} &L_{3}:=\{q\le 4673, ~q\ \textrm{prime}\} \cup \{5003,6007,7001,8009\},\\ &L_{4}:=\{q\le 1361, ~q\ \textrm{prime}\} \cup \{1409\}. \end{align*}的完全帽的最小尺寸.我们的研究和结果可以推测这些上界对所有的$q$都成立.
英文摘要：
In this paper we present and analyze computational results concerning small complete caps in the projective spaces $\mathrm{PG}(N,q)$ of dimension $N=3$ and $N=4$ over the finite field of order $q$. The results have been obtained using randomized greedy algorithms and the algorithm with fixed order of points (FOP). The computations have been done in relatively wide regions of $q$ values; such wide regions are not considered in literature for $N=3,4$. The new complete caps are the smallest known. Basing on them, we obtained new upper bounds on $t_2(N,q)$, the minimum size of a complete cap in $\mathrm{PG}(N,q)$, in particular, \begin{align*} &t_{2}(N,q)<\sqrt{N+2}\cdot q^{\frac{N-1}{2^{\vphantom{H}}}}\sqrt{\ln q},\quad q\in L_{N},\quad N=3,4,\\ &t_{2}(N,q)<\left(\sqrt{N+1}+\frac{1.3}{\ln (2q)}\right)q^{\frac{N-1}{2^{\vphantom{H}}}}\sqrt{\ln q},\quad q\in L_{N},\quad N=3,4, \end{align*} where \begin{align*} &L_{3}:=\{q\le 4673, ~q\ \textrm{prime}\} \cup \{5003,6007,7001,8009\},\\ &L_{4}:=\{q\le 1361, ~q\ \textrm{prime}\} \cup \{1409\}. \end{align*} Our investigations and results allow to conjecture that these bounds hold for all $q$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
