In a projective plane $Π_{q}$ (not necessarily Desarguesian) of order $q$, a point subset $\mathcal{S}$ is saturating (or dense) if any point of $Π_{q}\setminus \mathcal{S}$ is collinear with two points in $\mathcal{S}$. Modifying an approach of [31], we proved the following upper bound on the smallest size $s(2,q)$ of a saturating set in $Π_{q}$: \begin{equation*} s(2,q)\leq \sqrt{(q+1)\left(3\ln q+\ln\ln q +\ln\frac{3}{4}\right)}+\sqrt{\frac{q}{3\ln q}}+3. \end{equation*} The bound holds for all q, not necessarily large.
By using inductive constructions, upper bounds on the smallest size of a saturating set in the projective space $\mathrm{PG}(N,q)$ with even dimension $N$ are obtained.
All the results are also stated in terms of linear covering codes.
Further results on binary linear covering codes and saturating sets in projective spaces
Kharkevich Institute for Information Transmission Problems Russian Academy of Sciences（信息传输问题哈尔维奇研究所 俄罗斯科学院）；Department of Mathematics and Computer Science, University of Perugia（佩鲁贾大学数学与计算机科学系）；Independent（独立研究者）
Upper bounds on the length function for covering codes with covering radius $R$ and codimension $tR+1$
长度函数$\ell_q(r,R)$是具有余维（冗余）$r$和覆盖半径$R$的$ q $元线性码的最小长度。在这项工作中，新的上界$\ell_q(tR+1,R)$得到以下形式：\begin{equation*} \begin{split} &(a)~\ell_q(r,R)\le cq^{(r-R)/R}\cdot\sqrt[R]{\ln q},~ R\ge3,~r=tR+1,~t\ge1, &\phantom{(a)~} q\text{ is an arbitrary prime power},~c\text{ is independent of }q. \end{split} \end{equation*} \begin{equation*} \begin{split} &(b)~\ell_q(r,R)< 3.43Rq^{(r-R)/R}\cdot\sqrt[R]{\ln q},~ R\ge3,~r=tR+1,~t\ge1, &\phantom{(b)~} q\text{ is an arbitrary prime power},~q\text{ is large enough}. \end{split} \end{equation*}在文献中，对于$q=(q')^R$与$q'$一个素数的权力，较小的上界是已知的，但是，当$q$是一个任意的素数的权力，本文的界限是优于已知的。对于$…
Tables, bounds and graphics of the smallest known sizes of complete arcs in the plane $\mathrm{PG}(2,q)$ for all $q\le160001$ and sporadic $q$ in the interval $[160801\ldots 430007]$
在投影平面$\mathrm{PG}(2,q)$中，我们收集区域\begin{align*} &\mbox{all } q\le160001,~~ q \mbox{ prime power};\\ &Q_{4}=\{34 \mbox{ sporadic }q'\mbox{s in the interval }[160801\ldots430007], \mbox{ see Table 3}\}. \end{align*}的完整弧的最小已知尺寸。对于$q\le160001$，弧尺寸的收集在考虑所有素数幂的弧的意义上是完整的。这证明了$\mathrm{PG}(2,q)$，特别是\begin{align*} t_{2}(2,q)&<0.998\sqrt{3q\ln q}<1.729\sqrt{q\ln q}&\mbox{ for }&&7&\le q\le160001;~~(1) \\ t_{2}(2,q)&<\sqrt{q}\ln^{0.7295}q&\mbox{ for }&&109&\le q\le160001;~~(2)\\ t_{2}(2,q)&<\sqrt{q}\ln^{c_{up}(q)}q,~~c_{up}(q)=\frac{0.27}{\ln q}+0.7,&\mbox{ for }&&19&\le q\le160001;~~(3)\\ t_{2}(2,q)&<0.6\s…
Tables, bounds and graphics of sizes of complete arcs in the plane $\mathrm{PG}(2,q)$ for all $q\le321007$ and sporadic $q$ in $[323761\ldots430007]$ obtained by an algorithm with fixed order of points (FOP)
在作者以前的工作中，提出了一个逐步算法FOP，它使用任意固定顺序的点在投影平面$\mathrm{PG}(2,q)$。在每一步中，该算法将不位于弧的二等分线上的固定顺序的第一点添加到当前弧。该算法是基于直观的假设，$\mathrm{PG}(2,q)$包含足够数量的相对较小的完整的弧。此外，在以前的文件中，它表明，在$\mathrm{PG}(2,q)$的点的顺序的类型是不相关的。$\mathrm{PG}(2,q)$中的完备lexiarc是通过算法FOP使用点的字典序获得的完备弧。在这项工作中，我们收集并分析了以下区域中的完整lexiarcs的大小：\begin{align*}& \textbf{all } q\le321007,~ q \mbox{ prime power}; & 15 \mbox{ sporadic $q$'s in the interval }[323761\ldots430007], \mbox{ see (1.10)}. \end{align*}在工作[9]中，收集了所有$q\leq160001$，$q$素数幂的$\mathrm{PG}(2,q)$中完整弧的最小已知大小。完整的弧的大小，收集在这项工作中，在[9]，提供了以下上界的最小尺寸$t_{2}(2,q)$的一个完整的弧在投影平面$\mathrm{PG}(2,q)$：\begin{align*} t_{2}…
