For questions about Combinatorial design theory, a part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of balance and/or symmetry. The theory has applications in the area of the design ...

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Choosing sets of vectors on a complex sphere

Consider a complex $t$ dimensional unit sphere. Can we have $t$ sets of $2^t$ vectors $v_{ij}\in \Bbb C^t$ on the sphere where $i=1$ to $t$ and $j=1$ to $2^t$ on this with inner products satisfying ...
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Terminology - variant of a hypergraph

In a hypergraph, we have vertices $V$ and hyperedges $H$, where each hyperedge is a subset of $V$. Suppose that we would like the hyperedges to be (ordered) tuples, rather than subsets. Does this ...
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Steiner Triple Systems block clique

Given a Steiner Triple System (STS) of order $v$, one can build its graph in the following way: each vertex is a block, and two verticies are adjacent if their blocks have nonempty intersection. Thr ...
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1answer
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balanced incomplete designs

My textbook said a balanced design with covalency 0 is a complete design. I don't understand this, because $$\begin{gather} \text{set of varieties}=\{v_1,v_2,v_3\}\\ B_1 = \{v_1\},\\ B_2 = ...
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Number of blocks in this design $(V,\mathcal{B})$ $|V|=16$ and $\mathcal{B}$ has size $4$

Let $(V,\mathcal{B})$ be a design in which $|V|=16$, each block in $\mathcal{B}$ has size $4$, and each pair of points occurs in precisely one block. How many blocks are in this design? Try 1: ...
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Symmetric Balanced Incomplete Block Design Example with (56,7,1)

Can someone give an example of Symmetric Balanced Incomplete Block design with (56,7,1). That is v=b=56,k=r=7, $\Lambda = 1$.
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Find a (16,6,1) Balanced incomplete block design (BIBD)

I'm trying to find a balanced incomplete block design with the 16 items and $\lambda= 1$. I've calculated (using these defenitions) that a BIBD with 8 blocks and 6 items per block should be possible. ...
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41 views

Decomposing $K_v - K_u$ into Hamilton paths where $v = u^2 - u + 1$.

A decomposition of a graph $G$ into subgraphs $H$ is a collection of graphs all isomorphic to $H$ which are edge-disjoint in $G$ and together cover all the edges of $G$. Let $u \geq 1$ and $v = u^2 ...
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$24$ people in groups of $3$ where everyone meets exactly once at the end of some number of rounds

I was presented with this problem at work. Say you have $24$ people and $8$ tables in a room. You want to set people at these tables in groups of three such that during each new round (where people ...
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Out of all combinations (n,k), largest set such that each combination overlaps with others by d or less.

This problem is relevant to determining the number of discriminable combinations of components in a sensory perception task. Suppose that there are N components to choose from, and we are only ...
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Configuration analogues of projective spaces?

In a configuration, each point is incident to the same number of lines and each line is incident to the same number of points. The Fano plane is a configuration, with 3 points on each line, and 3 ...
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Question related to designs where t= 2

1)Let D be a 2−(v,k,λ) design with b blocks and r blocks through every point. Let B be any block. How to show that the number of blocks that meet B is at least $k(r−1)^2/[(k−1)(λ−1)+(r−1)]$ 2) How to ...
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Combinatorial designs give triangulations of complete graphs

I recently attended a talk on combinatorial design theory. The speaker mentioned briefly that the Fano plane, and other designs give rise to triangulations of complete graphs (the Fano plane gives a ...
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finding confounded effects

In a (3$^3$,3$^2$) design, how to find the effects confounded given the key block (0,0,0),(0,1,2) and (1,0,1)? I have completed the key block. ...
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Symmetric 2-Designs

We just proved that for any symmetric 2-design (or Symmetric BIBDs as most literature puts it) with parameters $(v,k,\lambda)$, any two blocks intersects at exactly $\lambda$ points. Our lecturer ...
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42 views

Upper bound for constant weight code L(n,d,w), with n=128, d=4

I would like to find an upper bound: L(n,d,w) <= f(n,d,w) for a constant weight code L(n,d,w), where w is the maximum weight, d is the Hamming distance between codes, and n is the code length. I ...
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A question involving Partial Steiner Triple Systems

I've been given the following question, which I think I've completed, but I just wanted to check whether what I've said is valid. Suppose that a PSTS(23) with a $K_5$ leave is constructed using ...
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A question regarding a combinatorial design.

I've been given the following question, and it almost seems too simple, so I'm not really too sure whether I'm just trying to overthink things. Let $B_0$ be a block of a $(v,k,1)$-design $(X, ...
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A proof regarding a symmetric $(v, k, \lambda)$-design.

I've been asked to prove the following; Consider a symmetric $(v,k,\lambda)$-design where $2k < v$. Prove that $2\lambda < k$. Now, being a symmetric design, I know that $v = b$, $r = k$, ...
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Estimate time it takes the minimum mean cycle cancelling algorithm to converge

This particular algorithm solves the circulation problem, equivalent to the minimum-capacitated flow. My question rather than only from this particular algorithm, but for combinatorial solutions in ...
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network design: why can't an almost satisfied proper function violated by all given sets?

I'm reading a book about (survivable) network design and i have a problem understanding a lemma. Given an undirected graph G and $V(G)$ its nodes and $E(G)$ its edges. The book defines a proper ...
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Counting the 1s in each row of the incidence matrix of a 2-design

Consider the $2 - (4t-1, 2t, t)$ design where $t$ is an odd number and $A$ is the incidence matrix. I suspect that the number of elements with value $1$ in each row of $A$ is equal to $2t$ but I can't ...
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Intersection of blocks of the symmetric BIBD $PG(d,q)$

The definition of a Balance Incomplete block design $(v,k,\lambda)$-BIBD can be found here. It is a well known fact (also see the link above) that every two blocks of a symmetric $(v,k,\lambda)$-BIBD ...
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Collection of subsets of $S$ where any $t$ have union equal to $S$ but any $t-1$ do not.

This recent question had me thinking about a generalization. Suppose we have a set of $n$ elements $S$. Suppose that we can assign the elements of $S$ to $b$ subsets $\{B_1,\ \cdots,\ B_b\}$ of $k$ ...
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45 views

Name for a type of combinatorial design?

Let $X$ be a ground set, and consider a collection $\mathscr{S}$ of subsets of $X$, $\mathscr{S} = \{S_1, \dots, S_n\}$. We would like to find a collection $\mathscr{S}'$ with the property that for ...
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Why doesnt a (43, 43, 7, 7, 1)-design exist according to the conditions?

I have tested this using the necessary conditions for a BIBD and it's giving me a green light but I know this isn't a design. Why not?