kb/data/en.wikipedia.org/wiki/Block_design-1.md

---
title: "Block design"
chunk: 2/6
source: "https://en.wikipedia.org/wiki/Block_design"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T09:49:03.148970+00:00"
instance: "kb-cron"
---

obtained from counting for a fixed x the triples (x, y, B) where x and y are distinct points and B is a block that contains them both. This equation for every x also proves that r is constant (independent of x) even without assuming it explicitly, thus proving that the condition that any x in X is contained in r blocks is redundant and r can be computed from the other parameters.
The resulting b and r must be integers, which imposes conditions on v, k, and λ. These conditions are not sufficient as, for example, a (43,7,1)-design  does not exist.
The order of a 2-design is defined to be n = r − λ.  The complement of a 2-design is obtained by replacing each block with its complement in the point set X.  It is also a 2-design and has parameters v′ = v, b′ = b, r′ = b − r, k′ = v − k, λ′ = λ + b − 2r.  A 2-design and its complement have the same order.
A fundamental theorem, Fisher's inequality, named after the statistician Ronald Fisher, is that b ≥ v in any 2-design.
A rather surprising and not very obvious (but very general) combinatorial result for these designs is that if points are denoted by  any arbitrarily chosen set of equally or unequally spaced numerics, there is no choice of such a set which can make all block-sums (that is, sum of all points in a given block) constant. For other designs such as partially balanced incomplete block designs this may however be possible. Many such cases are discussed in. However, it can also be observed trivially for the magic squares or magic rectangles which can be viewed as the partially balanced incomplete block designs.

=== Examples ===
The unique (6,3,2)-design (v = 6, k = 3, λ = 2) has 10 blocks (b = 10) and each element is repeated 5 times (r = 5). Using the symbols 0 − 5, the blocks are the following triples:

012     013     024     035     045     125     134     145     234     235.
and the corresponding incidence matrix (a v×b binary matrix with constant row sum r and constant column sum k) is:

  
    
      
        
          
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    {\displaystyle {\begin{pmatrix}1&1&1&1&1&0&0&0&0&0\\1&1&0&0&0&1&1&1&0&0\\1&0&1&0&0&1&0&0&1&1\\0&1&0&1&0&0&1&0&1&1\\0&0&1&0&1&0&1&1&1&0\\0&0&0&1&1&1&0&1&0&1\\\end{pmatrix}}}
  

One of four nonisomorphic (8,4,3)-designs has 14 blocks with each element repeated 7 times. Using the symbols 0 − 7 the blocks are the following 4-tuples:

0123    0124    0156    0257    0345    0367    0467    1267    1346    1357    1457    2347    2356    2456.
The unique (7,3,1)-design is symmetric and has 7 blocks with each element repeated 3 times. Using the symbols 0 − 6, the blocks are the following triples:

013     026     045     124     156     235     346.
This design is associated with the Fano plane, with the elements and blocks of the design corresponding to the points and lines of the plane. Its corresponding incidence matrix can also be symmetric, if the labels or blocks are sorted the right way:

  
    
      
        
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    {\displaystyle \left({\begin{matrix}1&1&1&0&0&0&0\\1&0&0&1&1&0&0\\1&0&0&0&0&1&1\\0&1&0&1&0&1&0\\0&1&0&0&1&0&1\\0&0&1&1&0&0&1\\0&0&1&0&1&1&0\end{matrix}}\right)}
  

== Symmetric 2-designs (SBIBDs) ==
The case of equality in Fisher's inequality, that is, a 2-design with an equal number of points and blocks, is called a symmetric design. Symmetric designs have the smallest number of blocks among all the 2-designs with the same number of points.
In a symmetric design r = k holds as well as b = v, and, while it is generally not true in arbitrary 2-designs, in a symmetric design every two distinct blocks meet in λ points. A theorem of Ryser provides the converse.  If X is a v-element set, and B is a v-element set of k-element subsets (the "blocks"), such that any two distinct blocks have exactly λ points in common, then (X, B) is a symmetric block design.
The parameters of a symmetric design satisfy

  
    
      
        λ
        (
        v
        −
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        )
        =
        k
        (
        k
        −
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        )
        .
      
    
    {\displaystyle \lambda (v-1)=k(k-1).}
  

This imposes strong restrictions on v, so the number of points is far from arbitrary.  The Bruck–Ryser–Chowla theorem gives necessary, but not sufficient, conditions for the existence of a symmetric design in terms of these parameters.
The following are important examples of symmetric 2-designs:

=== Projective planes ===

Finite projective planes are symmetric 2-designs with λ = 1 and order n > 1. For these designs the symmetric design equation becomes:

  
    
      
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        −
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        =
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        (
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        )
        .
      
    
    {\displaystyle v-1=k(k-1).}