Partitions of the 8‐dimensional vector space over GF(2)

Let V = V(n, q) denote the vector space of dimension n over GF(q). A set of subspaces of V is called a partition of V if every nonzero vector in V is contained in exactly one subspace of V. Given a partition 𝒫 of V with exactly a_i subspaces of dimension i for 1≤i≤n, we have , and we call the n‐tuple (a_n, a_{n − 1}, …, a₁) the type of 𝒫. In this article we identify all 8‐tuples (a₈, a₇, …, a₂, 0) that are the types of partitions of V(8, 2). © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 462–474, 2010     

Reconfiguration of subspace partitions

Let $q$ be a fixed prime power and let $V\left(n,q\right)$ denote a vector space of dimension $n$ over the Galois field with $q$ elements. A subspace partition (also called “vector space partition”) of $V\left(n,q\right)$ is a collection of subspaces of $V\left(n,q\right)$ with the property that every nonzero element of $V\left(n,q\right)$ appears in exactly one of these subspaces. Given positive integers $a,b,n$ such that $1\le a<b<n$, we say a subspace partition of $V\left(n,q\right)$ has type $a^x{b}^y$ if it is composed of $x$ subspaces of dimension $a$ and $y$ subspaces of dimension $b$. Let $c=\text{gcd}\left(a,b\right)$. In this paper, we prove that if $b$ divides $n$, then one can (algebraically) construct every possible subspace partition of $V\left(n,q\right)$ of type $a^x{b}^y$ whenever $y\ge \left(q^e?1\right)∕\left(q^b?1\right)$, where $0\le e<ab∕c$ and $n\equiv e\left(\mathrm{mod}ab∕c\right)$. Our construction allows us to sequentially reconfigure batches of $\left(q^a?1\right)∕\left(q^c?1\right)$ subspaces of dimension $b$ into batches of $\left(q^b?1\right)∕\left(q^c?1\right)$ subspaces of dimension $a$. In particular, this accounts for all numerically allowed subspace partition types $a^x{b}^y$ of $V\left(n,q\right)$ under some additional conditions, for example, when $e=b$.     

Some infinite families of 2‐designs from PG (n,q)

In this paper, we establish the existence of some infinite families of 2‐designs from ‐dimensional projective geometry , which admit ‐dimensional projective special linear group as their flag‐transitive automorphism group.     

Partitioning 2‐Edge‐Colored Ore‐Type Graphs by Monochromatic Cycles

Consider a graph G on n vertices satisfying the following Ore‐type condition: for any two nonadjacent vertices x and y of G, we have . We conjecture that if we color the edges of G with two colors then the vertex set of G can be partitioned to two vertex disjoint monochromatic cycles of distinct colors. In this article, we prove an asymptotic version of this conjecture.     

奇异伪辛群作用下子空间轨道的长度

设F_q是q个元素的有限域,q是2的幂,F_q~(2ν+δ+l)是F_q上2ν+δ+l维行向量空间,Ps_(2ν+δ+l,2ν+δ)(F_q)是F_q上级数为2ν+δ+l而秩为2ν+δ的伪辛群.F_q~(2ν+δ+l)在Ps_(2ν+δ+l,2ν+δ)(F_q)的作用下划分成一些子空间轨道Μ(m,2s+τ,s,∈,k;2ν+δ,2ν+δ).采用矩阵初等行变换的方法,给出轨道Μ(m,2s+τ,s,∈,k;2ν+δ,2ν+ε)的长度.     

New lower bounds on the size of (n,r)‐arcs in PG(2,q)

An $\left(n,r\right)$‐arc in $\text{PG}\left(2,q\right)$ is a set of $n$ points such that each line contains at most $r$ of the selected points. It is well known that $\left(n,r\right)$‐arcs in $\text{PG}\left(2,q\right)$ correspond to projective linear codes. Let $m_r\left(2,q\right)$ denote the maximal number $n$ of points for which an $\left(n,r\right)$‐arc in $\text{PG}\left(2,q\right)$ exists. In this paper we obtain improved lower bounds on $m_r\left(2,q\right)$ by explicitly constructing $\left(n,r\right)$‐arcs. Some of the constructed $\left(n,r\right)$‐arcs correspond to linear codes meeting the Griesmer bound. All results are obtained by integer linear programming.     

算子子空间的渐近自反性和遗传性

研究了算子子空间的渐近自反性问题，渐近自反子空间的遗传斯近自反性以及某些单个算子的渐近自反性．我们也讨论了投影网类的浙近自反性。     

3‐Spontaneous Emission Error Designs from PSL(2, q) or PGL(2, q)

Quantum jump codes are quantum codes that correct errors caused by quantum jumps. A spontaneous emission error design (SEED) was introduced by Beth et al. in 2003 to construct quantum jump codes. In this paper, we study the existence of 3‐SEEDs from PSL(2, q) or PGL(2, q). By doing this, a large number of 3‐ SEEDs are derived for prime powers q and all .     

Blocking sets in PG(2, q n ) from cones of PG(2n, q)

Let Ω and be a subset of Σ = PG(2n−1,q) and a subset of PG(2n,q) respectively, with Σ ⊂ PG(2n,q) and . Denote by K the cone of vertex Ω and base and consider the point set B defined by

Generalized quadrangles admitting PSL(2,q) × PSL(2,q)

In this paper, we classify the generalized quadrangles of order (s,t), s ≠ 1 ≠ t, which admit the natural action of PSL(2,s) × PSL(2,s) on a subGQ of order (s,1). This generalizes a recent result of J. De Kaey and H. Van Maldeghem 3 , by whom the classification was obtained for the case s = t. © 2005 Wiley Periodicals, Inc. J Combin Designs.     

Small Point Sets that Meet All Generators of W(2n+1,q)

Let W(2n+1,q), n1, be the symplectic polar space of finite order q and (projective) rank n. We investigate the smallest cardinality of a set of points that meets every generator of W(2n+1,q). For q even, we show that this cardinality is q ⁿ⁺¹+q ^{n–1, and we characterize all sets of this cardinality. For q odd, better bounds are known.     

A Note on q‐Covering Designs in PG(5, q)

A construction of q‐covering designs in PG(5, q) is given, providing an improvement on the upper bound of the q‐covering number .     

On the mutually non isomorphic ℓp(ℓq) spaces

We extend a result of Pe?czyński showing that {?_p(?_q): 1 ≤ p, q ≤ ∞} is a family of mutually non isomorphic Banach spaces. Some results on complemented subspaces of ?_p(?_q) are also given.     

On a Problem of Judicious k‐Partitions of Graphs

For positive integers and m , let be the smallest integer such that for each graph G with m edges there exists a k‐partition in which each contains at most edges. Bollobás and Scott showed that . Ma and Yu posed the following problem: is it true that the limsup of tends to infinity as m tends to infinity? They showed it holds when k is even, establishing a conjecture of Bollobás and Scott. In this article, we solve the problem completely. We also present a result by showing that every graph with a large k‐cut has a k‐partition in which each vertex class contains relatively few edges, which partly improves a result given by Bollobás and Scott.     

On the Action of the Groups GL(n+1, q), PGL(n + 1, q), SL(n + 1, q) and PSL(n + 1, q) on PG(n, q t)

Let q be a prime power and let n ≥ 0, t ≥ 1 be integers. We determine the sizes of the point orbits of each of the groups GL(n + 1, q), PGL(n + 1, q), SL(n + 1, q) and PSL(n + 1, q) acting on PG(n, q ^t) and for each of these sizes (and groups) we determine the exact number of point orbits of this size.     

On Perfect 2‐Colorings of Johnson Graphs J(v, 3)

We study the perfect 2‐colorings (also known as the equitable partitions into two parts or the completely regular codes with covering radius 1) of the Johnson graphs . In particular, we classify all the realizable quotient matrices of perfect 2‐colorings for odd v. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 232–252, 2013