design (original) (raw)

A τ-(ν,κ,λ) design, aka τ-design or block design, is an incidence structure (𝒫,ℬ,ℐ) with

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  |𝒫|=ν points in all,
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  |𝒫B|=κ points in each block B, and such that
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  any set T⊆𝒫 of |T|=τ points occurs as subset T⊆𝒫B in exactly λ blocks.

The numbers τ,ν,κ,l⁢a⁢m⁢b⁢d⁢a are called the parameters of a design. They are often called t, v, k, λ (in mixed Latin and Greek alphabets) by some authors.

Given parameters τ,ν,κ,l⁢a⁢m⁢b⁢d⁢a, 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 𝒫B.

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  0-designs (τ=0) are allowed.
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  1-designs (τ=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 τ-(ν,κ,λ) design implies also being an ι-(ν,κ,λι) design for every 0≤ι≤τ (on the same ν points and with the same block size κ), with λι given by λτ=λ and recursively

from which we get the number of blocks as

Being a 0-design says nothing more than all blocks having the same size. As soon as we have τ≥1 however we also have a 1-design, so the number λ1=|ℬP| of blocks per point P is constant throughout the structure. Note now

which is also evident from their interpretation.

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

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  τ=0 implies no more than that,
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A more elaborate “lambda calculus” (pun intended) can be introduced as follows. Let I⊆P and O⊆P with |I|=ι 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 ι and o, provided ι+o≤τ. Call this number λιo. It satisfies a kind of reverse Pascal triangle likerecursion

that starts off for o=0 with λι0=λι. An important quantity (for designs with τ≥2) is the order λ11=λ10-λ20=λ1-λ2.

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

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  A square design aka symmetric design is one where τ=2and |𝒫|=|ℬ|, now also |𝒫B|=|ℬP|. Here the dual is also a square design.

Note that for τ≥3 no designs exist with |𝒫|=|ℬ| other than trivial ones (where any κ=ν-1 points form a block).