All Questions
Tagged with combinatorial-designs co.combinatorics
91
questions
3
votes
1
answer
408
views
Ranks of higher incidence matrices of designs
In 1978 Doyen, Hubaut and Vandensavel proved that if $S$ is a Steiner triple system $S(2,3,v)$ then the $GF(2)$ rank of its incidence matrix $N$ is
$$
Rk_{2}(N)=v-(d_{p}+1),
$$
where $d_{p}$ is the ...
- 6,832
3
votes
3
answers
678
views
Orthogonal Latin Square 6*6
I need to make remarks about Tarry's Proof for the nonexistence of 6x6 Latin Squares as part of my final exam for a class I'm in. Problem is, I can't find it ANYWHERE on the internet. I can only find ...
- 39
3
votes
1
answer
125
views
On the half-skew-centrosymmetric Hadamard matrices
Definition 1: A Hadamard matrix is an $n\times n$ matrix $H$ whose entries are either $1$ or $-1$ and whose rows are mutually orthogonal.
Definition 2: A matrix $A$ is half-skew-centrosymmetric if ...
- 264
3
votes
1
answer
134
views
Mutually orthogonal Latin hypercubes
A $d$-dimensional Latin hypercube with side length $n$ is a $d$-dimensional array with $n$ symbols such that along any line parallel to an axis, each symbol appears exactly once. Let us call a $(n,d)$ ...
- 645
3
votes
1
answer
117
views
On the existence of a certain graph/hypergraph pair
Let $V$ be a finite set, $G$ a simple graph with vertex set $V$, and $H$ a hypergraph (i.e., set of subsets) with vertex set $V$ satisfying the following three conditions:
each pair of elements of $V$...
- 21.7k
3
votes
1
answer
267
views
Lower bounding the maximum size of sets in a set family with union promise
The following problem has come up while working on the relationship between certificate and randomized decision tree complexities of boolean functions. However, I think it is of interest by itself and ...
3
votes
0
answers
39
views
Existence of finite 3-dimensional hyperbolic balanced geometry
Together with @TarasBanakh we faced the problem described in the title. Let me start with definitions.
A linear space is a pair $(S,\mathcal L)$ consisting of a set $S$ and a family $\mathcal L$ of ...
- 441
3
votes
0
answers
51
views
Cliques in Incomplete block designs
I'm interested in inequalities that guarantee the presence of cliques in incomplete block designs. Here's the set-up:
I have an incidence structure $(V, B)$ which is an incomplete block design: $V$ is ...
- 11.1k
3
votes
0
answers
132
views
Linear combinations of special matrices
I am a hobby computer scientist and I have a problem to which I am searching an efficient algorithm.
Given an integer n, we want to combine some square input-matrices of size n in a way that is ...
- 73
3
votes
0
answers
129
views
Graeco-Latin squares and outer-automorphisms
It is well known that $n=6$ is the only number greater than two in which there is no Graeco-Latin square of order $n$. It is also well known that $n=6$ is the only number greater than two in which ...
- 2,471
3
votes
0
answers
91
views
what is the largest real orthogonal design in $n$ variables?
A real orthogonal design in $n$ variables is an $m \times n$ matrix with
entries from the set $\pm x_1,\pm x_2,\cdots,\pm x_n$ that satisfies :
$$ A A^T = (x_1^2 + x_2^2 + \cdots x_n^2) I_m $$
...
- 451
3
votes
0
answers
121
views
A question on the behavior of intersections of certain block design
Let $[d]$ be a universe and $S_1, \dots, S_m$ be an $(\ell, a)$-design over $[d]$ which means that:
$\forall i \in [m], S_i \subseteq [d], |S_i|=\ell$.
$\forall i \neq j \in [m]$, $|S_i \cap S_j| \...
- 143
2
votes
3
answers
299
views
Constructions of $2-(v,3,3)$-designs
I am looking for ways to construct an infinite family of designs with parameters $2-(v,3,3)$ and apart from some doubling-type recursive constructions (such as in this paper) I haven't found anything ...
- 6,832
2
votes
2
answers
277
views
Resolvable designs from projective space
Resolvable designs are block designs with the additional property that the blocks can be partitioned into partitions of the points. It is easy to see that lines in affine space form a resolvable ...
- 7,525
2
votes
2
answers
762
views
Minimally intersecting subsets of fixed size
The question I have, is how to generate the following collection of subsets:
Given a set $S$ of size $n$. I want to find a sequence of $k$ subsets of fixed size $m$, $0<m<n$, such that at each ...
- 145
2
votes
1
answer
59
views
Constructing Group Divisible Designs - Algorithms?
I am starting my research on group divisible designs this year and I wonder if there are any algorithms/software that help with constructions.
Thank you
- 29
2
votes
2
answers
211
views
Minimal number of blocks in a $(n,n/2,\lambda)$ block design
A $(n,n/2,\lambda)$ block-design is a family $A_1,...,A_K$ of subsets of $[n]$ such that $|A_i|=n/2$ and for every $1 \leq i < j \leq n$ it holds that
$\#\{1 \leq k \leq K : i,j \in A_k \} = \...
- 21
2
votes
1
answer
201
views
Hitting sets (aka covers aka transversals) of Steiner triple systems
Does there exist a constant $c$ so that the lines of every Steiner
triple system on $v$ points can be covered by $cv$ points?
That is if $D \in STS(v)$ with point set $T=\{1,2,\ldots,v\}$ then ...
- 6,832
2
votes
1
answer
128
views
Distinguishing points by sets of given size
The problem is:
Given a finite set $X$ with size $x$ and let $B$ denote a family of $k$-element subsets of $X$, called blocks. What is the smallest possible number $n$ of blocks such that every ...
- 9,282
2
votes
1
answer
164
views
Lower bounds on cardinality of a union of blocks in a design
Let $D$ be a $(v,k,\lambda)$-design (repeated blocks are allowed). I would like to get a lower bound on the cardinality of the union of $s$ blocks. A naive application of inclusion-exclusion gives $sk-...
- 6,832
2
votes
1
answer
207
views
Has this kind of design been studied before?
Consider a design $(X,\mathcal{B})$, satisfying:
Each block in $\mathcal{B}$ has the same size
The intersection of every two blocks has the same size
Of course, it is easy to find many examples of ...
- 53
2
votes
1
answer
302
views
Existence of Steiner system designs given $n,k,t$
I am familiar with the recent Keevash paper here which proves that given some $t,n,k,\lambda$ then provided standard divisibility conditions hold, and $n$ is suitably large, there exists a $t-(n,k,\...
2
votes
1
answer
424
views
Popular elements in cross-intersecting families
Let $\mathcal{T}$ and $\mathcal{S}$ be two families of subsets of $[n]$ such that for all $T_i\in \mathcal{T}$ and $S_j\in \mathcal{S}$,
$|T_i \cap S_j| \neq\emptyset$
$|T_i| , |S_j| \leq t = O(\log(...
- 21
2
votes
0
answers
193
views
Non-uniform Ray-Chaudhuri-Wilson (generalized Fisher's inequality)
A $t$-design on $v$ points with block size and index $\lambda$ is a collection $\mathcal{B}$ of subsets of a set $V$ with $v$ elements satisfying the following properties:
(a) every $B\in\mathcal{B}$ ...
- 19.1k
2
votes
0
answers
67
views
Point sets with tangents through every point
Let $D=(P,L)$ be either a $(v,k,\lambda)$-design or a near-linear space (or, more generally, any incidence structure with "points" and sets of points which are called "blocks" or "lines") and let $S \...
- 6,832
1
vote
5
answers
348
views
Pairwise balanced designs with $r=\lambda^{2}$
A while ago I asked how to construct an infinite family of $(v,b,r,k,\lambda)$-designs satisfying $r=\lambda^{2}$ and got very good answers from Yuichiro Fujiwara and Ken W. Smith.
Now I'd like to up ...
- 6,832
1
vote
1
answer
448
views
The Symmetry of Steiner System S(5,8,24)
The group of automorphisms of S(5,8,24), M_{24}, is 5-transitive.
Other than Symmetric groups are there any other 5-transitive groups?
If not, would it be correct to say S(5,8,24) is the most ...
- 25
1
vote
1
answer
123
views
On the existence of symmetric matrices with prescribed number of 1's on each row
We are considering the following problem:
Given an integer $n$ and a sequence of integers $r_i,\ 1\le i\le n$, with $0\le r_i\le n-1$ does there exists a symmetric matrix $A$ such that the diagonal ...
- 13
1
vote
1
answer
134
views
3-partition of a special set
$S_5$ is a set consisting of the following 5-length sequences $s$: (1) each digit of $s$ is $a$, $b$, or $c$; (2) $s$ has and only has one digit that is $c$.
$T_5$ is a set consisting of the following ...
- 25
1
vote
1
answer
795
views
Known results on cyclic difference sets
Is there any infinite family of $v$ for which all the $(v,k,\lambda)$-cyclic difference sets with $k-\lambda$ a prime power coprime to $v$ have been determined?
A subset $D=\{a_1,\ldots,a_k\}$ of $\...
- 767
1
vote
1
answer
68
views
One question about nega-cyclic Hadamard matrices
Let $n$ be a multiple of $4$, is there any $n \times n$ negacyclic Hadamard matrix? If yes - how to construct it? If no - why?
Here an $n \times n$ nega-cyclic matrix is a square matrix of the form:
\...
- 264
1
vote
1
answer
154
views
Number of points in an intersecting linear hypergraph
I first asked the question below at math.stackexchange.com ( https://math.stackexchange.com/questions/920442/number-of-points-in-an-intersecting-linear-hypergraph ) but somebody suggested I ask it in ...
- 44.5k
1
vote
0
answers
80
views
Bounds for smallest non-trivial designs
Given $s>t\ge 2$, let $N(s,t)$ be the smallest integer $n>s$ such that there exists an “$(n;s;t;1)$-design” (i.e., a collection of $s$-subsets $e_1,\dots,e_m$ of $[n]:=\{1,\dots,n\}$, such that ...
- 2,874
1
vote
0
answers
48
views
Optimal choice of points to maximize majorities in a $t-(v,k,\lambda)$ design
Let us consider a design $\mathcal{D} = (V,\mathcal{B})$ with points in $V$ and blocks in $\mathcal{B}$. I am interested in the special case of a $t-(v,k,\lambda)$ design for $k=3$, i.e., all blocks ...
- 143
1
vote
1
answer
298
views
Covering designs where $v$ is linear in $k$
A $(v,k,t)$ covering design is a collection of $k$-subsets of $V=\{1,\ldots,v\}$ chosen so that any $t$-subset of $V$ is contained in (or "covered by") at least one $k$-set in the collection. ...
0
votes
1
answer
157
views
"JigSaw Puzzle" on Set Family
One of my research problem can be reduced to a question of the following form
Given a set family $\mathcal{F}$ of $[n]$ , such that every element of $[n]$ lies in exactly $K$ sets in $\mathcal{F}$, ...
- 344
0
votes
1
answer
238
views
a block design question: Does every special 1-design admit a partition which respects enough of the blocks?
Is it possible to show that every 1-design $D$ with $\lambda=4,k=4$ on $v$ points (for $v$ that is a multiple of $3$) contain some 1-design $Q$ with $\lambda=1,k=3$ on $v$ points such that every block ...
- 509
0
votes
1
answer
49
views
Vector version of balanced incomplete block designs
I am interested in finding out what is known about the following generalization of balanced incomplete block designs (BIBDs):
"What is the maximum size of a collection $B$ of $v$-dimensional unit ...
- 103
0
votes
1
answer
111
views
All $2$-designs arising from the action of the affine linear group on the field of prime order
Let $p$ be a prime and $\mathbb{Z}_p$ denote as usual the field of order $p$. Let $AL(p)$ be the affine linear group $\{x\mapsto ax+b \;|\; a\in \mathbb{Z}_p\setminus \{0\}, b\in\mathbb{Z}_p\}$. For a ...
- 3,708
0
votes
0
answers
90
views
Steiner-like systems with large edges and many intersections
Let $l\geq 3$ be an integer. Is there $n\in\mathbb{N}$ and a hypergraph $H=(\{1,\ldots,n\},E)$ with the following properties?
for all $e\in E$ we have $|e| \geq l$
$e_1\neq e_2 \in E \implies |e_1 \...
- 44.5k
-1
votes
1
answer
373
views
Number of blocks in a t-(v,k,l) design with empty intersection with a given set U [closed]
Question
Given a $t-(v,k,\lambda)$ design $(X,\mathcal{B})$ and a set $U\subset X$ with $|U|=u\leq t$, what is the number of blocks $B\in\mathcal{B}$ such that $B\cap U=\emptyset$?
The answer is: $\...
- 1