On Blocking Sets of Affine Spaces
Ara Aleksanyan, Mihran Papikian
本文研究在三阶有限域上的仿射空间中求与所有直线相交所需的最少点数的问题。我们还考虑了在二阶域上的一个仿射空间中求与所有二维仿射子空间相交的最小点数的问题。
We consider the problem of finding the minimal number of points required to intersect all lines in an affine space over the finite field of order 3. We also consider the problem of finding the minimal number of points required to intersect all two dimensional affine subspaces in an affine space over the field of order 2.
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A new lower bound for the size of an affine blocking set
仿射平面中的分块集是一组点$B$，使得每条线至少包含$B$的一个点。最著名的下界块集在任意（非Desklesian）仿射平面是在20世纪80年代由Bruen和Silverman。本文改进了这一结果，证明了$q$阶仿射平面的块集$q\geq 25$至少包含$q+\lfloor\sqrt{q}\rfloor+3$点.
Constructing minimal blocking sets using field reduction
给出了$\mathrm{PG}(n-1,q^t)$空间中$(k-1)$ -空间的最小阻塞集的一个构造. $(k-1)$是$q^t$阶有限域$\mathbb{F}_{q^t}$上的$(n-1)$维射影空间.该构造依赖于在$\mathrm{PG}(n-1,q^t)$的{ \em场缩减}表示中使用阻塞锥，扩展了线性阻塞集的公知构造。这种结构的灵感来自于Mazzocca，Polverino和Storme（{ \em的MPS-建设}）的超平面的最小阻塞集的建设，我们表明，对于一个合适的选择的阻塞锥平面阻塞集，我们得到更大的阻塞集比从平面阻塞集\cite{pol}。进一步证明了在$\mathrm{PG}(nt-1,q)$中对关于$(nt-t-1)$ -空间的极小块集进行域约化，可以得到关于$\mathrm{PG}PG(n-1,q^t)$中超平面的极小块集.最后，我们将这些结构的线性猜想小的最小块集。我们证明了，如果一个小的极小阻塞集构造的MPS-结构，它是Rédei型，而一个小的极小阻塞集产生于我们的锥结构当且仅当它是线性的。
On MPS construction of blocking sets in projective spaces: a generalization
本文利用$s\leq n-2$维的子空间给出了$PG(r,q^n)$的分块集的MPS构造的推广。通过这种构造，我们确定了$PG(3,q^6)$中的一个新的非平面例子。
A note on strong blocking sets and higgledy-piggledy sets of lines
本文研究了有限射影空间中的强阻塞集，证明了某些并集不能构成强阻塞集从而得到新的大小下界，并在特定条件下构造了呈杂乱排列的直线子集以生成强阻塞集。
Sets of generators blocking all generators in finite classical polar spaces
引入了有限经典极空间的生成块集。这些集合是最大部分点差的推广。本文证明了极空间Q（2n，q），Q-（2n+1，q）和H（2n，q^2）的极小集的一个特征，即顶点为包含在极空间中的子空间，底为秩为2的极空间中的生成块集的锥.
Blocking sets in small finite linear spaces II
本文的内容已被纳入数学CO/0308288。
The main purpose of this paper is to find double blocking sets in $\mathrm{PG}(2,q)$ of size less than $3q$, in particular when $q$ is prime. To this end, we study double blocking sets in $\mathrm{PG}(2,q)$ of size $3q-1$ admitting at least two $(q-1)$-secants. We derive some structural properties of these and show that they cannot have three $(q-1)$-secants. This yields that one cannot remove six points from a triangle, a double blocking set of size $3q$, and add five new points so that the resulting set is also a double blocking set. Furthermore, we give constructions of minimal double block…
On certain blocking sets and the minimum weight of the code of generalised polygons
本文通过刻画广义多边形中的阻塞集，推导出关联矩阵生成的码在最小重量及最小重量码字方面的新结果。
Solution to Bishnoi's conjecture on minimal t-fold blocking sets of maximal size
Bishnoi证明了如果素数幂阶的射影平面中的极小t-重阻塞集具有最大尺寸，则它要么是一个射影平面减一点，要么是Baer子平面的补，要么是一个酉集。本文证明了这一猜想。
On the linearity of higher-dimensional blocking sets
PG（n，q）（q = pt，p素数）中的极小k-阻塞集B是指PG（n，q）中小于3（qk + 1）/2个点的集合，使得每个（n-k）维空间至少包含B的一个点，且B的任何真子集都不满足这一性质。线性猜想指出PG（n，q）中的所有极小k-块集在Fq的子域Fpe上是线性的。除了少数情况外，这个猜想仍然是开放的。本文证明了PG（n，pt）中k-分块集的线性猜想的证明，其中指数e和pe \geq 7，只要n的一个值至少为2k就足够了.进一步，我们证明了PG（2，q）中的小最小阻塞集的线性性蕴涵了PG（n，pt）中的小最小k-阻塞集的线性性，其指数为e，pe \geq t/e + 11.
On the smallest sets blocking simple perfect matchings in a convex geometric graph
本文给出了完全凸几何图中在$2m$顶点上阻塞所有简单完美匹配的最小集合的完全刻画。特别地，我们证明了所有这些集合都是具有特殊结构的毛虫图，并且它们的总数是$m \cdot 2^{m-1}$。
A proof of the linearity conjecture for k-blocking sets in PG(n, p3), p prime
本文证明了PG（n，q3），q = p^h，h >= 1，p素，p >=7中与每个（n-k）-空间相交于1（mod q）点的极小k-块集是线性的.作为推论，这个结果表明PG（n，p^3），p素数，p >=7中的所有极小k-分块集都是FP-线性的，证明了在PG（n，p3），p素数，p >= 7的情况下的线性猜想（见[7]）。
Small 3-fold blocking sets in $\mathrm{PG}(2,p^n)$
本文构造了PG(2,p^n)中猜想大小的3-重阻集，通过三个雷德伊型线性阻集的不相交并集实现，并位于特定的轨道上。
We investigate the upper chromatic number of the hypergraph formed by the points and the $k$-dimensional subspaces of $\mathrm{PG}(n,q)$; that is, the most number of colors that can be used to color the points so that every $k$-subspace contains at least two points of the same color. Clearly, if one colors the points of a double blocking set with the same color, the rest of the points may get mutually distinct colors. This gives a trivial lower bound, and we prove that it is sharp in many cases. Due to this relation with double blocking sets, we also prove that for $t\leq \frac38p+1$, a small …
Blocking Sets of Index Three
在这篇笔记中，我们将提供的各种声明，已在文献中所作的关于阻塞集的索引3的证明。我们的目的是澄清什么是已知的这些集的特征。具体地说，我们提供了PG（2，q）中索引为3的所有Rédei阻塞集的构造，并显式地确定了PG（2，7）中索引为3的所有阻塞集。
We prove that a minimal $t$-fold blocking set in a finite projective plane of order $n$ has cardinality at most \[\frac{1}{2} n\sqrt{4tn - (3t + 1)(t - 1)} + \frac{1}{2} (t - 1)n + t.\] This is the first general upper bound on the size of minimal $t$-fold blocking sets in finite projective planes and it generalizes the classical result of Bruen and Thas on minimal blocking sets. From the proof it directly follows that if equality occurs in this bound then every line intersects the blocking set $S$ in either $t$ points or $\frac{1}{2}(\sqrt{4tn - (3t + 1)(t - 1)} + t - 1) + 1$ points. We use th…
Flat-containing and shift-blocking sets in $F_2^r$
对于非负整数$r\ge d$，假设对于任何$v\in F_2^r$，都有一个$d$ -平坦通过$v$并包含在$C\cup\{v\}$中，那么子集$C\subset F_2^r$能有多小？等价地，假设对于任何$v\in F_2^r$，都有一个线性$d$ -子空间不被平移$B+v$非平凡地阻塞，那么子集$B\subset F_2^r$可以有多大？得到了一些上界和下界。
Blocking sets in chain geometries
本文给出了有限链几何的最小尺寸块集的一些基本结果。
A small minimal blocking set in PG(n,p^t), spanning a (t-1)-space, is linear
本文证明了在PG（n，p^t），p素，（t/e-1）维空间中，当p>5（t/e）-11时，指数为e的极小分块集是F_p^e-线性集.作为推论，我们得到了PG（n，p^t），p素，p> 5 t-11中所有跨越（t-1）维空间的极小块集都是F_p-线性的，从而证实了块集在这种特殊情况下的线性猜想.
Blocking sets in small finite linear spaces
我们分类所有有限线性空间上最多15点允许一个块集。在11个或更少的点上没有这样的空格，在12个点上有一个，在13个点上有一个，在14个点上有两个，在15个点上有五个。证明广泛使用的概念的一个点的重量在一个2色有限线性空间，以及区分最小和非最小2色有限线性空间。然后，我们使用这种分类得出一些结论，两个开放的问题上的2-着色的配置点。
Minimal codes are being intensively studied in last years. $[n,k]_{q}$-minimal linear codes are in bijection with strong blocking sets of size $n$ in $PG(k-1,q)$ and a lower bound for the size of strong blocking sets is given by $(k-1)(q+1)\leq n$. In this note we show that all strong blocking sets of length 9 in $PG(3,2)$ are the hyperbolic quadrics $Q^{+}(3,2)$.
On bisecants of Rédei type blocking sets and applications
我们使用多项式的技术，从他们的二等分信息Rédei型阻塞集的结构结果。我们将我们的结果应用到$PG(2,q)$的点集与几个奇割线。特别地，我们改进了Balister，Bollobás，Füredi和Thompson关于$PG(2,q)$中$(q+2)$ -集的奇割线个数的下界，并回答了Vandendriessche的一个相关的公开问题.我们证明了半椭圆的结构结果，并得出不存在大小为$q+3$的半椭圆时，3不除$q$和$q>5$。这扩展了Blokhuis将大小为$q+2$的半椭圆分类的结果，以及Bartoli将大小为$q+3$的半椭圆分类为$q\leq 17$的结果。在$q$甚至情况下，我们可以说更多的应用结果Szynyi和韦纳的稳定性集的甚至类型。对Gács和Weiner关于$(q+t,t)$ -弧型$(0,2,t)$的一个结果以及Ball，Blokhuis，Brouwer，Storme和Szynyi关于$GF(q)$上的函数决定小于$(q+3)/2$方向的一个结果的一部分，我们也得到了新的证明.
Strong blocking sets and their counterparts, minimal codes, attracted lots of attention in the last years. Combining the concatenating construction of codes with a geometric insight into the minimality condition, we explicitly provide infinite families of small strong blocking sets, whose size is linear in the dimension of the ambient projective spaces. As a byproduct, small saturating sets are obtained.
Groups with maximal irredundant covers and minimal blocking sets
令$n$为正整数。用$\mathrm{PG}(n,q)$表示$q$阶有限域$\mathbb{F}_q$上的$n$维射影空间. $\mathrm{PG}(n,q)$中的阻塞集是与$\mathrm{PG}(n,q)$的每个超平面具有非空交集的点的集合。一个阻塞集称为极小的，如果它的真子集都不是阻塞集。本文证明了如果$\mathrm{PG}(n_i,q)$包含$i\in\{1,2\}$的大小为$k_i$的最小阻塞集，则$\mathrm{PG}(n_1+n_2+1,q)$包含大小为$k_1+k_2-1$的最小阻塞集.这一结果是证明了一个结果的群体与最大的非冗余覆盖。
Outer Strong Blocking Sets
本文提出外部强阻塞集概念，研究其结构与大小界，改进强阻塞集最小大小的上界，并给出计算成本更低的小型强阻塞集几何构造。
Dominating sets in projective planes
通过建立有限射影平面的关联图的小控制集的稳定性结果，证明了控制集与块集和覆盖集之间的强相关性.我们的主要结果表明，如果$q>81$阶射影平面中的控制集小于$2q+2[\sqrt{q}]+2$（即，Baer子平面大小的两倍），则它包含一条直线的所有点，但可能只有一个点，或者包含通过一个点的所有线，但可能只有一条线。此外，我们完全刻画了大小至多为$2q+\sqrt{q}+1$的控制集。在Despermesian平面上，我们可以依靠关于阻塞集的强稳定性结果来证明，如果一个控制集足够小于3q，那么它由一个阻塞集和一个覆盖集的并集组成，除了几个点和线。
Blocking sets, minimal codes and trifferent codes
本文建立仿射阻塞集、强阻塞集、极小码与线性三相异码之间的等价关系，据此改进相关上下界，并给出更大的显式线性三相异码构造。
Classification theorem for strong triangle blocking arrangements
强三角形阻塞排列是指某些线段在具有一定相交性质的三角形中的几何排列。事实证明，它们与阻塞集密切相关。本文的目的是证明一个强三角块安排的分类定理。作为应用，我们得到了Ackerman，Buchin，Knauer，Pinchasi和Rote关于$n$点在一般位置不能被$n-1$点遮挡的结果的一个新证明，除非$n = 2,4$ .我们还推测一个极端的变化的阻塞点问题。
Plane curves giving rise to blocking sets over finite fields
本文研究有限域上平面曲线的有理点何时产生阻塞集，证明低次不可约曲线提供阻塞集，并利用数论工具构造特定次数的光滑曲线。
Minimal linear codes are algebraic objects which gained interest in the last twenty years, due to their link with Massey's secret sharing schemes. In this context, Ashikhmin and Barg provided a useful and a quite easy to handle sufficient condition for a linear code to be minimal, which has been applied in the construction of many minimal linear codes. In this paper, we generalize some recent constructions of minimal linear codes which are not based on Ashikhmin-Barg's condition. More combinatorial and geometric methods are involved in our proofs. In particular, we present a family of codes ar…
Blocking Coloured Point Sets
本文研究了平面上点之间的可见性问题。如果$x$在线段$\bar{vw}$的内部，则点$x$ \emph{blocks}为两点$v$和$w$。点集合$P$是\emph{$k$-blocked}，如果$P$中的每个点被分配$k$颜色之一，使得当且仅当$P$块$v$和$w$中的某个其他点时，不同的点$v,w\in P$被分配相同的颜色。本文的重点是猜想每个$k$ -阻塞集有界大小（作为$k$的函数）。文献中的结果表明，每个2-块集最多有3个点，每个3-块集最多有6个点。我们证明了每个4-阻塞集最多有12个点，并且这个界是紧的。事实上，我们对所有集合$\{n_1,n_2,n_3,n_4\}$进行排序，使得某个4块集合恰好在第$i$个颜色类中具有$n_i$点。在其他结果中，对于无穷多个$k$值，我们用$k^{1.79...}$点构造$k$ -阻塞集。
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 als…
Minimal additive codes and additive strong blocking sets
本文引入最小加法码的概念，通过建立与加法强阻塞集的一一对应关系，从几何角度研究其性质，并给出构造、存在性结果以及最小长度的上、下界和渐近界。
Blocking sets from a union of plane curves
本文受Erdős问题启发，利用固定次数几何不可约曲线的并集在有限域射影平面上构造阻塞集，并给出所需曲线数量的下界，证明结合算术几何与组合方法。
In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, which were recently introduced by Bonini and Borello, and then completely characterize minimal linear codes as cutting blocking sets. As a direct result, minimal projective codes of dimension $3$ and $t$-fold blocking sets with $t\ge 2$ in projective planes are identical objects. Some bounds on the parameters of minimal codes are derived from this characterization. This confirms a recent conjecture by Alfarano, Borello and Neri in [a geometric characterization of minimal codes and t…
Strong blocking sets and minimal codes from expander graphs
本文结合常数度扩张图与渐近好码，在有限射影空间中显式构造了大小为 $O(qk)$ 的强阻塞集，从而首次给出了满足 $n=O(qk)$ 的极小线性码显式构造，解决了该领域的主要开放问题。
Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. In this paper we prove an upper bound on the minimal length of minimal codes of dimension $k$ over the $q$-element Galois field which is linear in both $q$ and $k$, hence improve the previous superlinear bounds. This result determines the minimal length up to a small constant factor. We also improve the lower and upper bounds on the size of so called higgledy-piggledy line sets in projective spaces and apply these …
Zero blocking numbers of graphs with complexity results
本文研究图的零阻塞数，确定了两个图的并和联的零阻塞数、超立方体的所有最小零阻塞集，并给出树的零阻塞数的线性时间算法。
Explicit constructions of optimal blocking sets and minimal codes
本文提出了一种显式构造强s-阻集的方法，适用于有限射影空间，并由此得到最优的仿射阻集和s-极小码构造，方法基于expander图的构造，并引入了特定的超图结构。
Minimal rank-metric codes or, equivalently, linear cutting blocking sets are characterized in terms of the second generalized rank weight, via their connection with evasiveness properties of the associated $q$-system. Using this result, we provide the first construction of a family of $\mathbb{F}_{q^m}$-linear MRD codes of length $2m$ that are not obtained as a direct sum of two smaller MRD codes. In addition, such a family has better parameters, since its codes possess generalized rank weights strictly larger than those of the previously known MRD codes. This shows that not all the MRD codes …
On the code generated by the incidence matrix of points and k-spaces in PG(n, q) and its dual
本文研究了PG（n，q）中点与k维空间的关联矩阵生成的p元线性码Ck（n，q），q = ph，p素，h >= 1.对于k >= n/2，我们将权重小于2 q ^k的Ck（n，q）\Ck（n，q）的码字链接到k-分块集合。我们首先证明这样的k-分块集可以唯一地约简为最小k-分块集，并排除从小线性k-分块集产生的所有码字。对于k < n/2，我们给出了对k >= n/2有效的引理的反例。其次，我们研究了Ck（n，q）的对偶码，给出了其码字重量的一个下界，从而将Sachar [12]的结果推广到一般维数.
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…
We prove an analogue for homogeneous trees and certain affine buildings of a result of Bourgain on pinned distances in sets of positive density in Euclidean spaces. Furthermore, we construct an example of a non-homogeneous tree with positive Hausdorff dimension, and a subset with positive density thereof, in which not all sufficiently large (even) distances are realised.
Blocking subspaces with points and hyperplanes
本文研究有限射影几何中同时用点和超平面阻挡所有k-空间的最小集合问题，证明当k不等于(n-1)/2时最优解只含一种元素，而在临界情形下混合构造可以严格更小。
Classes and equivalence of linear sets in $PG(1,q^n)$
研究了$PG(1,q^n)$的秩为n的$\mathbb{F}_q$ -线性集的等价性问题，包括伴随簇、投影配置、$\mathbb{F}_q$ -Rédei型线性块集和MRD-码.
The study of symmetric configurations $v_3$ with block size 3 has a long and rich history. In this paper we consider two colouring problems which arise naturally in the study of these structures. The first of these is weak colouring, in which no block is monochromatic; the second is strong colouring, in which every block is multichromatic. The former has been studied before in relation to blocking sets. Results are proved on the possible sizes of blocking sets and we begin the investigation of strong colourings. We also show that the known $21_3$ and $22_3$ configurations without a blocking se…
Maximal integral point sets in affine planes over finite fields
受欧几里德平面上整点集的启发，我们考虑了有限域上仿射平面上的整点集。整数点集是有限域$\mathbb{F}_q$上仿射平面$\mathbb{F}_q^2$中的点的集合，其中每对点的形式定义的平方欧几里德距离是$\mathbb{F}_q$中的平方。证明了$\mathbb{F}_q$上的整点集也可以刻画为确定某些预定方向的仿射点集，这与Blokhuis的工作有关。此外，在一个重要的子情况整点集可以重述为团的佩利图的平方阶。本文给出了整点集自同构的新结果，并对$q\le 47$的$\mathbb{F}_q$上的极大整点集进行了分类.进一步给出了两类极大整点集，并证明了它们的极大性。
This work investigates the structure of rank-metric codes in connection with concepts from finite geometry, most notably the $q$-analogues of projective systems and blocking sets. We also illustrate how to associate a classical Hamming-metric code to a rank-metric one, in such a way that various rank-metric properties naturally translate into the homonymous Hamming-metric notions under this correspondence. The most interesting applications of our results lie in the theory of minimal rank-metric codes, which we introduce and study from several angles. Our main contributions are bounds for the p…
In this paper, we consider the following problem: given a connected graph $G$, can we reduce the domination number of $G$ by one by using only one edge contraction? We show that the problem is $\mathsf{NP}$-hard when restricted to $\{P_6,P_4+P_2\}$-free graphs and that it is $\mathsf{coNP}$-hard when restricted to subcubic claw-free graphs and $2P_3$-free graphs. As a consequence, we are able to establish a complexity dichotomy for the problem on $H$-free graphs when $H$ is connected.
Minimal blocking sets in PG(n,2) and covering groups by subgroups
本文证明了PG（n，2）的点集$B$是极小阻塞集当且仅当$B$是$<B>=PG(d,2)$的点集，且$B$是$PG(d,2)$的$d+2$点集，且它们的$d+1$不在同一超平面上.作为后一个结果的推论，我们证明了：如果$G$是有限2-群且$n$是正整数，则$G$允许$\mathfrak{C}_{n+1}$ -覆盖当且仅当$n$是偶数且$G\cong (C_2)^{n}$，其中，对于群$H$的$\mathfrak{C}_m$ -覆盖，我们表示$H$的极大子群的大小为$m$的集合$\mathcal{C}$，其集合-理论并是整个$H$，没有$\mathcal{C}$的真子集具有后一性质，极大子群的交是无核的.同样对于所有$n<10$，我们找到所有对$(m,p)$（$m>0$是整数，$p$是素数），对于这些对，存在$PG(m,p)$中大小为$n$的阻塞集$B$，使得$<B>=PG(m,p)$。
We describe the $\mathfrak sl_3$ non-elliptic webs in terms of convex sets in the affine building. Kuperberg defined the non-elliptic web basis in his work on rank-$2$ spider categories. Fontaine, Kamnitzer, Kuperberg showed that the $\mathfrak sl_3$ non-elliptic webs are dual to CAT(0) triangulated diskoids in the affine building. We show that each such triangulated diskoid is the intersection of the min-convex and max-convex hulls of a generic polygon in the building. Choosing a generic polygon from each of the components of the Satake fiber produces the non-elliptic web basis. The convex hu…
Maximal integral point sets in affine planes over finite fields
受欧几里德平面上整点集的启发，我们考虑了有限域上仿射平面上的整点集。整数点集是有限域$\mathbb{F}_q$上仿射平面$\mathbb{F}_q^2$中的点的集合，其中每对点的形式定义的平方欧几里德距离是$\mathbb{F}_q$中的平方。证明了$\mathbb{F}_q$上的整点集也可以刻画为确定某些预定方向的仿射点集，这与Blokhuis的工作有关。此外，在一个重要的子情况整点集可以重述为团的佩利图的平方阶。本文给出了整点集自同构的新结果，并对$q\leq 47$的$\mathbb{F}_q$上的极大整点集进行了分类.进一步给出了两类极大整点集，并证明了它们的极大性。
Prime Power Residues and Blocking Sets
本研究建立数论中素数模$q$次幂剩余问题与伽罗瓦几何中阻塞集的等价联系，证明该剩余性质在几何$q$等价下不变，并据此分类极小整数集且给出其大小界。
Affine Equivalence of Subsets of $\mathbb{F}_2^n$ via Venn Diagrams and Applications to Sidon Sets
本文利用仿射张成基构造划分集合的维恩图，证明维恩区域间保持基数的线性置换可完全刻画 $\mathbb{F}_2^n$ 子集的仿射等价性，并据此分类若干 Sidon 集。
In this note, we introduce a new approach to abstract ``synthetic'' projective lines. We discuss various aspects of our approach, and compare these aspects with the classical one. A number of intriguing questions arise. Amongst these aspects, we discuss geometric closures, and introduce automorphism-blocking sets. We also construct a number of (counter) examples in infinite cases.
Cameron-Liebler sets for maximal totally isotropic flats in classical affine spaces
本文研究经典仿射空间中极大全迷向平面集合上的Cameron-Liebler集，给出若干等价定义并得到分类结果。
In the $m$-dimensional affine space $AG(m,q)$ over the finite field $\mathbb{F}_q$ of odd order $q$, the analogous of the Euclidean distance gives rise to a graph $\mathfrak{G}_{m,q}$ where vertices are the points of $AG(m,q)$ and two vertices are adjacent if their (formal) squared Euclidean distance is a square in $\mathbb{F}_q$ (including the zero). In 2009, Kurz and Meyer made the conjecture that if $m$ is even then $\mathfrak{G}_{m,q}$ is a strongly regular graph. In this paper we prove their conjecture.
Avoiding intersections of given size in finite affine spaces AG(n,2)
研究 AG(n,2) 中仿射子空间与点集的交集大小，部分确定了不可避免的局部密度，构造了基于子空间规避集的点集，并改进了其上界。
In this paper we study the number of special directions of sets of cardinality divisible by $p$ on a finite plane of characteristic $p$, where $p$ is a prime. We show that there is no such a set with exactly two special directions. We characterise sets with exactly 3 special directions which answers a question of Ghidelli in negative. Further we introduce methods to construct sets of minimal cardinality that has exactly 4 special directions for small values of $p$.
A Simple Explicit Construction of Smaller Strong Blocking Sets
本文提出一种简单方法，通过级联(q-1)幂相交代数几何码构造显式强阻塞集与最小码，q≤11时性能优于Alon等人的构造，且无需依赖扩张器图。
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