---
title: "Block design"
chunk: 3/6
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"
---

Since k = r we can write the order of a projective plane as n = k − 1 and, from the displayed equation above, we obtain v = (n + 1)n + 1 = n2 + n + 1 points in a projective plane of order n.
As a projective plane is a symmetric design, we have b = v, meaning that b = n2 + n + 1 also. The number b is the number of lines of the projective plane. There can be no repeated lines since λ = 1, so a projective plane is a simple 2-design in which the number of lines and the number of points are always the same. For a projective plane, k is the number of points on each line and it is equal to n + 1.  Similarly, r = n + 1 is the number of lines with which a given point is incident.
For n = 2 we get a projective plane of order 2, also called the Fano plane, with v = 4 + 2 + 1 = 7 points and 7 lines. In the Fano plane, each line has n + 1 = 3 points and each point belongs to n + 1 = 3 lines.
Projective planes are known to exist for all orders which are prime numbers or powers of primes. They form the only known infinite family (with respect to having a constant λ value) of symmetric block designs.

=== Biplanes ===
A biplane or biplane geometry is a symmetric 2-design with λ = 2; that is, every set of two points is contained in two blocks ("lines"), while any two lines intersect in two points. They are similar to  finite projective planes, except that rather than two points determining one line (and two lines determining one point), two points determine two lines (respectively, points). A biplane of order n is one whose blocks have k = n + 2 points; it has v = 1 + (n + 2)(n + 1)/2 points (since r = k).
The 18 known examples are listed below.

(Trivial) The order 0 biplane has 2 points (and lines of size 2;  a 2-(2,2,2) design); it is two points, with two blocks, each consisting of both points. Geometrically, it is the digon.
The order 1 biplane has 4 points (and lines of size 3; a 2-(4,3,2) design); it is the complete design with v = 4 and k = 3. Geometrically, the points are the vertices of a tetrahedron and the blocks are its faces.
The order 2 biplane is the complement of the Fano plane: it has 7 points (and lines of size 4; a 2-(7,4,2)), where the lines are given as the complements of the (3-point) lines in the Fano plane.
The order 3 biplane has 11 points (and lines of size 5; a 2-(11,5,2)), and is also known as the Paley biplane after Raymond Paley; it is associated to the Paley digraph of order 11, which is constructed using the field with 11 elements, and is the Hadamard 2-design associated to the size 12 Hadamard matrix; see Paley construction I.
Algebraically this corresponds to the exceptional embedding of the projective special linear group PSL(2,5) in PSL(2,11) – see projective linear group: action on p points for details.
There are three biplanes of order 4 (and 16 points, lines of size 6; a 2-(16,6,2)). One is the Kummer configuration. These three designs are also Menon designs.
There are four biplanes of order 7 (and 37 points, lines of size 9; a 2-(37,9,2)).
There are five biplanes of order 9 (and 56 points, lines of size 11; a 2-(56,11,2)).
Two biplanes are known of order 11 (and 79 points, lines of size 13; a 2-(79,13,2)).
Biplanes of orders 5, 6, 8 and 10 do not exist, as shown by the Bruck-Ryser-Chowla theorem.

=== Hadamard 2-designs ===
An Hadamard matrix of size m is an m × m matrix H whose entries are ±1 such that HH⊤  = mIm, where H⊤ is the transpose of H and Im is the m × m identity matrix. An Hadamard matrix can be put into standardized form (that is, converted to an equivalent Hadamard matrix) where the first row and first column entries are all +1. If the size m > 2 then m must be a multiple of 4.
Given an Hadamard matrix of size 4a in standardized form, remove the first row and first column and convert every −1 to a 0. The resulting 0–1 matrix M is the incidence matrix of a symmetric 2-(4a − 1, 2a − 1, a − 1) design called an Hadamard 2-design. 
It contains 
  
    
      
        4
        a
        −
        1
      
    
    {\displaystyle 4a-1}
  
 blocks/points; each contains/is contained in 
  
    
      
        2
        a
        −
        1
      
    
    {\displaystyle 2a-1}
  
 points/blocks. Each pair of points is contained in exactly 
  
    
      
        a
        −
        1
      
    
    {\displaystyle a-1}
  
 blocks.
This construction is reversible, and the incidence matrix of a symmetric 2-design with these parameters can be used to form an Hadamard matrix of size 4a.

== Resolvable 2-designs ==
A resolvable 2-design is a BIBD whose blocks can be partitioned into sets (called parallel classes), each of which forms a partition of the point set of the BIBD. The set of parallel classes is called a resolution of the design.
If a 2-(v,k,λ) resolvable design has c parallel classes, then b  ≥ v + c − 1.
Consequently, a symmetric design can not have a non-trivial (more than one parallel class) resolution.
Archetypical resolvable 2-designs are the finite affine planes.  A solution of the famous 15 schoolgirl problem is a resolution of a 2-(15,3,1) design.