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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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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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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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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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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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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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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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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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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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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Applications of Steiner Systems

I am looking for mathematical applications of Steiner systems (and other mathematical designs). The wikipedia article says that: This area is one of the oldest parts of combinatorics, such as in ...
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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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Non Crossing Partitions

How can I prove that if $\phi$ is an automorphism of the poset $NC(n)$, there exists an $\alpha \in S_{n}$ such that $\phi(\pi) = \alpha\pi$ for every $\pi\in NC(n)$ and the group of automorphisms of ...