Binzhou Xia
The covering radius of a subset $C$ of the symmetric group $\mathrm{S}_n$ is the maximal Hamming distance of an element of $\mathrm{S}_n$ from $C$. This note determines the covering radii of the finite projective general linear groups. It turns out that the covering radius of $\mathrm{PGL}_2(q)$ is $q-2$ if $q$ is even, and is $q-3$ if $q$ is odd.
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Here is a simplified proof that every sharply transitive subset of $\mathrm{PGL}_2(K)$ is a coset of a subgroup.
Reduced Power Graphs of $\mathrm{PGL}_3(\mathbb{F}_q)$
本文研究 $\mathrm{PGL}_3(\mathbb{F}_q)$ 的约化幂图，确定了其连通分量个数及各分量直径的精确值。
Given a group $G$, let us connect two non-identity elements by an edge if and only if one is a power of another. This gives a graph structure on $G$ minus identity, called the reduced power graph. It is conjectured by Akbari and Ashrafi that if a non-abelian finite simple group has a connected reduced power graph, then it must be an alternating group. In this paper, we shall give a complete description of when the reduced power graphs of $\mathrm{PGL}_n(\mathbb{F}_q)$ are connected for all $q$ and all $n\geq 3$. In particular, the conjectured by Akbari and Ashrafi is false. We shall also provi…
A closure operator on the subgroup lattice of $\mathrm{GL}(n,q)$ and $\mathrm{PGL}(n,q)$ in relation to the zeros of the Möbius function
本文在PGL(V)和GL(V)的子群格上定义闭包算子，用以刻画Möbius函数非零的子群，并对满足特定不变子空间格条件的指数为m的闭子群个数建立了仅依赖于m的多项式上界。
In this paper we study combinatorial invariants of the equivalence classes of pencils of cubics on $\mathrm{PG}(1,q)$, for $q$ odd and $q$ not divisible by 3. These equivalence classes are considered as orbits of lines in $\mathrm{PG}(3,q)$, under the action of the subgroup $G\cong \mathrm{PGL}(2,q)$ of $\mathrm{PGL}(4,q)$ which preserves the twisted cubic $\mathcal{C}$ in $\mathrm{PG}(3,q)$. In particular we determine the point orbit distributions and plane orbit distributions of all $G$-orbits of lines which are contained in an osculating plane of $\mathcal{C}$, have non-empty intersection w…
Let $\mathrm{PG}(3,q)$ be the projective space of dimension three over the finite field with $q$ elements. Consider a twisted cubic in $\mathrm{PG}(3,q)$. The structure of the point-plane incidence matrix in $\mathrm{PG}(3,q)$ with respect to the orbits of points and planes under the action of the stabilizer group of the twisted cubic is described. This information is used to view generalized doubly-extended Reed-Solomon codes of codimension four as asymptotically optimal multiple covering codes.
A $q$-covering design $\mathbb{C}_q(n, k, r)$, $k \ge r$, is a collection $\mathcal X$ of $(k-1)$-spaces of $\mathrm{PG}(n-1, q)$ such that every $(r-1)$-space of $\mathrm{PG}(n-1, q)$ is contained in at least one element of $\mathcal X$ . Let $\mathcal{C}_q(n, k, r)$ denote the minimum number of $(k-1)$-spaces in a $q$-covering design $\mathbb{C}_q(n, k, r)$. In this paper improved upper bounds on $\mathcal{C}_q(2n, 3, 2)$, $n \ge 4$, $\mathcal{C}_q(3n + 8, 4, 2)$, $n \ge 0$, and $\mathcal{C}_q(2n,4,3)$, $n \ge 4$, are presented. The results are achieved by constructing the related $q$-coveri…
Boundedness and Separation Between Induced and Non-Induced Covering Numbers
本文研究诱导与非诱导覆盖数之间的有界性，针对不同性质的图类，在240种组合中为219种确定了绑定函数的存在性，并证明遗传客类在三个覆盖数上总有绑定函数。
We show that for any $n$ and $q$, the number of real conjugacy classes in $\mathrm{PGL}(n, \mathbb{F}_q)$ is equal to the number of real conjugacy classes of $\mathrm{GL}(n, \mathbb{F}_q)$ which are contained in $\mathrm{SL}(n, \mathbb{F}_q)$, refining a result of Lehrer, and extending the result of Gill and Singh that this holds when $n$ is odd or $q$ is even. Further, we show that this quantity is equal to the number of real conjugacy classes in $\mathrm{PGU}(n, \mathbb{F}_q)$, and equal to the number of real conjugacy classes of $\mathrm{U}(n, \mathbb{F}_q)$ which are contained in $\mathrm{…
A maximal minor $M$ of the Laplacian of an $n$-vertex Eulerian digraph $Γ$ gives rise to a finite group $\mathbb{Z}^{n-1}/\mathbb{Z}^{n-1}M$ known as the sandpile (or critical) group $S(Γ)$ of $Γ$. We determine $S(Γ)$ of the generalized de Bruijn graphs $Γ=\mathrm{DB}(n,d)$ with vertices $0,\dots,n-1$ and arcs $(i,di+k)$ for $0\leq i\leq n-1$ and $0\leq k\leq d-1$, and closely related generalized Kautz graphs, extending and completing earlier results for the classical de Bruijn and Kautz graphs. Moreover, for a prime $p$ and an $n$-cycle permutation matrix $X\in\mathrm{GL}_n(p)$ we show that $…
On the Computational Complexity of Local and Global Covering Numbers
研究了局部和全局覆盖数的计算复杂性，证明了在特定图类下确定局部覆盖数的NP难性，并给出了一个计算全局覆盖数的高效图类示例。
In this paper we fix 7 types of undirected graphs: paths, paths with prescribed endvertices, circuits, forests, spanning trees, (not necessarily spanning) trees and cuts. Given an undirected graph $G=(V,E)$ and two "object types" $\mathrm{A}$ and $\mathrm{B}$ chosen from the alternatives above, we consider the following questions. \textbf{Packing problem:} can we find an object of type $\mathrm{A}$ and one of type $\mathrm{B}$ in the edge set $E$ of $G$, so that they are edge-disjoint? \textbf{Partitioning problem:} can we partition $E$ into an object of type $\mathrm{A}$ and one of type $\mat…
Block-transitive $3$-$(v,k,1)$ designs associated with alternating groups
本文证明：若块传递的 $3$-$(v,k,1)$ 设计具有几乎单自同构群且基座为交错群，则该设计必为唯一的 $3$-$(10,4,1)$ 设计，且群为 $\rm{PGL}(2,9)$、$\rm{M}_{10}$ 或 $\rm{Aut}(\rm{A}_6)$，并是旗传递的。
The length function $\ell_q(r,R)$ is the smallest length of a $ q $-ary linear code of codimension $r$ and covering radius $R$. In this work we obtain new constructive upper bounds on $\ell_q(r,R)$ for all $R\ge4$, $r=tR$, $t\ge2$, and also for all even $R\ge2$, $r=tR+\frac{R}{2}$, $t\ge1$. The new bounds are provided by infinite families of new covering codes with fixed $R$ and increasing codimension. The new bounds improve upon the known ones. We propose a general regular construction (called ``Line+Ovals'') of a minimal $ρ$-saturating $((ρ+1)q+1)$-set in the projective space $\mathrm{PG}(2ρ…
Kronecker classes, normal coverings and chief factors of groups
本文修正了普雷杰关于有限群A-覆盖子群的定理证明，探讨了克罗内克类与群结构的关系。
Given a set $S$ of $v \ge 2$ symbols, and integers $k \ge t \ge 2$ and $N \ge 1$, an $N \times k$ array $A \in S^{N \times k}$ is an $(N; t, k, v)$-covering array if all sequences in $S^t$ appear as rows in every $N \times t$ subarray of $A$. These arrays have a wide variety of applications, driving the search for small covering arrays. The covering array number, $\mathrm{CAN}(t,k,v)$, is the smallest $N$ for which an $(N; t,k,v)$-covering array exists. In this paper, we combine probabilistic and linear algebraic constructions to improve the upper bounds on $\mathrm{CAN}(t,k,v)$ by a factor of…
Nearly Sharp Bounds for Lattice Coverings by Convex Bodies
该研究针对n维凸体的格覆盖密度，通过垂直-水平构造等方法，将其最坏情形增长指数优化为近线性，改进了此前O(n²)的通用界，确定了最优多项式增长指数。
Given a simple graph $G$, a set $C \subseteq V(G)$ is a neighborhood cover set if every edge and vertex of $G$ belongs to some $G[v]$ with $v \in C$, where $G[v]$ denotes the subgraph of $G$ induced by the closed neighborhood of the vertex $v$. Two elements of $E(G) \cup V(G)$ are neighborhood-independent if there is no vertex $v\in V(G)$ such that both elements are in $G[v]$. A set $S\subseteq V(G)\cup E(G)$ is neighborhood-independent if every pair of elements of $S$ is neighborhood-independent. Let $ρ_{\mathrm n}(G)$ be the size of a minimum neighborhood cover set and $α_{\mathrm n}(G)$ of …
New upper bound for lattice covering by spheres
本文证明了存在密度为 $O(n\ln^{1.858}n)$ 的等半径球格覆盖，改进了 Rogers 的 $O(n\ln^{2.047}n)$ 上界。
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