Editing Incidence structure (section)

===Block designs===
{{main|Block design}}
A (general) block design is a set {{mvar|X}} together with a [[Family of sets|family {{mvar|F}} of subsets]] of {{mvar|X}} (repeated subsets are allowed). Normally a block design is required to satisfy numerical regularity conditions. As an incidence structure, {{mvar|X}} is the set of points and {{mvar|F}} is the set of lines, usually called ''blocks'' in this context (repeated blocks must have distinct names, so {{mvar|F}} is actually a set and not a multiset). If all the subsets in {{mvar|F}} have the same size, the block design is called ''uniform''. If each element of {{mvar|X}} appears in the same number of subsets, the block design is said to be ''regular''. The dual of a uniform design is a regular design and vice versa.

====Example: Fano plane====
Consider the block design/hypergraph given by:
<math display="block">\begin{align}
P &= \{1,2,3,4,5,6,7\}, \\[2pt]
L &= \left\{ \begin{array}{ll}
  \{1,2,3\}, & \{1,4,5\}, \\
  \{1,6,7\}, & \{2,4,6\}, \\
  \{2,5,7\}, & \{3,4,7\}, \\
  \{3,5,6\} \end{array} \right\}.
\end{align}</math>

This incidence structure is called the [[Fano plane]]. As a block design it is both uniform and regular.

In the labeling given, the lines are precisely the subsets of the points that consist of three points whose labels add up to zero using [[nim addition]]. Alternatively, each number, when written in [[binary number|binary]], can be identified with a non-zero vector of length three over the [[GF(2)|binary field]]. Three vectors that generate a [[linear subspace|subspace]] form a line; in this case, that is equivalent to their vector sum being the zero vector.