Block Design (original) (raw)

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An incidence system (v,k, lambda, r, b) in which a set X of v points is partitioned into a family A of b subsets (blocks) in such a way that any two points determinelambda blocks with k points in each block, and each point is contained in r different blocks. It is also generally required that k<v, which is where the "incomplete" comes from in the formal term most often encountered for block designs, balanced incomplete block designs (BIBD).

The five parameters are not independent, but satisfy the two relations

vr=bk (1)
lambda(v-1)=r(k-1). (2)

A BIBD is therefore commonly written as simply (v, k,lambda), since b and r are given in terms of v, k, and lambda by

A BIBD is called symmetric if b=v (or, equivalently, r=k).

Writing X={x_i}_(i=1)^v and A={A_j}_(j=1)^b, then the incidence matrix of the BIBD is given by the v×b matrix M defined by

m_(ij)={1 if x_i in A_j; 0 otherwise. (5)

This matrix satisfies the equation

MM^(T)=(r-lambda)I+lambdaJ, (6)

where I is a v×v identity matrix and J is the v×v unit matrix (Dinitz and Stinson 1992).

Examples of BIBDs are given in the following table.

block design (v, k,lambda)
affine plane (n^2, n, 1)
Fano plane (7, 3, 1)
Hadamard design symmetric (4n+3, 2n+1, n)
projective plane symmetric (n^2+n+1, n+1, 1)
Steiner triple system (v, 3, 1)
unital (q^3+1,q+1, 1)

See also

Affine Plane, Design, Fano Plane, Hadamard Design, Parallel Class, Projective Plane, Resolution, Resolvable,Steiner Triple System, Symmetric Block Design, Unital

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References

Dinitz, J. H. and Stinson, D. R. "A Brief Introduction to Design Theory." Ch. 1 in Contemporary Design Theory: A Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson). New York: Wiley, pp. 1-12, 1992.Ryser, H. J. "The (b,v,r,k,lambda)-Configuration." §8.1 in Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., pp. 96-102, 1963.

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Block Design

Cite this as:

Weisstein, Eric W. "Block Design." FromMathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BlockDesign.html

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