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

  • |𝒫|=ν points in all,

  • |𝒫B|=κ points in each block B, and such that

  • any set T𝒫 of |T|=τ points occurs as subset T𝒫B in exactly λ blocks.

The numbers τ,ν,κ,lambda 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,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 𝒫B.

  • 0-designs (τ=0) are allowed.

  • 1-designs (τ=1) are known as tactical configurations.

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

  • 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 0ιτ (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 τ1 however we also have a 1-design, so the number λ1=|P| of blocks per point P is constant throughout the structureMathworldPlanetmath. Note now


which is also evident from their interpretationMathworldPlanetmath.

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

A more elaborate “lambda calculusMathworldPlanetmath” (pun intended) can be introduced as follows. Let IP and OP 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+oτ. Call this number λιo. It satisfies a kind of reverse Pascal triangleMathworldPlanetmath like recursion


that starts off for o=0 with λι0=λι. An important quantity (for designs with τ2) is the order λ11=λ10-λ20=λ1-λ2.

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

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

Note that for τ3 no designs exist with |𝒫|=|| other than trivial ones (where any κ=ν-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 τ-design
Synonym BIBD
Defines block
Defines simple design
Defines square design
Defines symmetric design
Defines tactical configuration
Defines balanced incomplete block design