# design

A $\tau$-$(\nu,\kappa,\lambda)$ design, aka $\tau$-design or , is an incidence structure $(\mathcal{P},\mathcal{B},\mathcal{I})$ with

• $|\mathcal{P}|=\nu$ points in all,

• $|\mathcal{P}_{B}|=\kappa$ points in each block $B$, and such that

• any set $T\subseteq\mathcal{P}$ of $|T|=\tau$ points occurs as subset $T\subseteq\mathcal{P}_{B}$ in exactly $\lambda$ blocks.

The numbers $\tau,\nu,\kappa,lambda$ are called the parameters of a design. They are often called $t$, $v$, $k$, $\lambda$ (in mixed Latin and Greek alphabets) by some authors.

Given parameters $\tau,\nu,\kappa,lambda$, there may be several non-isomorphic designs, or no designs at all.

Designs need not be simple (they can have repeated blocks), but they usually are (and don’t) in which case $B$ can again be used as synonym for $\mathcal{P}_{B}$.

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0-designs ($\tau=0$) are allowed.

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1-designs ($\tau=1$) are known as tactical configurations.

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2-designs are called balanced incomplete block designs or BIBD.

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3, 4, 5… -designs have all been studied.

Being a $\tau$-$(\nu,\kappa,\lambda)$ design implies also being an $\iota$-$(\nu,\kappa,\lambda_{\,\iota})$ design for every $0\leq\iota\leq\tau$ (on the same $\nu$ points and with the same block size $\kappa$), with $\lambda_{\,\iota}$ given by $\lambda_{\,\tau}=\lambda$ and recursively

 $\lambda_{\,\iota}\,=\;{\nu-\iota\over\kappa-\iota}\,\lambda_{\,\iota+1}$

from which we get the number of blocks as

 $\lambda_{0}\,=\;{\nu!\,/\,(\nu-\tau)!\over\kappa!\,/\,(\kappa-\tau)!}\;=\;{\nu% \choose\tau}\bigg{/}{\kappa\choose\tau}$

Being a 0-design says nothing more than all blocks having the same size. As soon as we have $\tau\geq 1$ however we also have a 1-design, so the number $\lambda_{1}=|\mathcal{B}_{P}|$ of blocks per point $P$ is constant throughout the structure. Note now

 $\lambda_{0}\,\kappa\;=\;\lambda_{1}\,\nu$

which is also evident from their interpretation.

As an example: designs (simple designs) with $\kappa=2$ are multigraphs (simple graphs), now

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$\tau=0$ implies no more than that,

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$\tau=1$ gives , and

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$\tau=2$ gives .

A more elaborate “lambda calculus” (pun intended) can be introduced as follows. Let $I\subseteq P$ and $O\subseteq P$ with $|I|=\iota$ and $|O|=o$. The number of blocks $B$ such that all the points of $I$ are inside $B$ and all the points of $O$ are outside $B$ is independent of the choice of $I$ and $O$, only depending on $\iota$ and $o$, provided $\iota+o\leq\tau$. Call this number $\lambda_{\iota}^{o}$. It satisfies a kind of reverse Pascal triangle like recursion

 $\lambda_{\iota}^{o}\,=\,\lambda_{\iota+1}^{o}+\lambda_{\iota}^{o+1}$

that starts off for $o=0$ with $\lambda_{\iota}^{0}=\lambda_{\iota}$. An important quantity (for designs with $\tau\geq 2$) is the order $\lambda_{1}^{1}=\lambda_{1}^{0}-\lambda_{2}^{0}=\lambda_{1}-\lambda_{2}$.

Finally, the dual of a design can be a design but need not be.

• A square design aka symmetric design is one where $\tau=2$ and $|\mathcal{P}|=|\mathcal{B}|$, now also $|\mathcal{P}_{B}|=|\mathcal{B}_{P}|$. Here the dual is also a square design.

Note that for $\tau\geq 3$ no designs exist with $|\mathcal{P}|=|\mathcal{B}|$ other than trivial ones (where any $\kappa=\nu-1$ points form a block).

 Title design Canonical name Design Date of creation 2013-03-22 19:14:09 Last modified on 2013-03-22 19:14:09 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 5 Author CWoo (3771) Entry type Definition Classification msc 62K10 Classification msc 51E30 Classification msc 51E05 Classification msc 05B25 Classification msc 05B07 Classification msc 05B05 Synonym block design Synonym tau-design Synonym $\tau$-design Synonym BIBD Defines block Defines simple design Defines square design Defines symmetric design Defines tactical configuration Defines balanced incomplete block design