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# Encyclopaedia of DesignTheory: Nets

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# Nets

A net of order n and degree r is an incidence structure (whose elements are called points and lines) having the following properties:
• There are n² points and nr lines.
• Each line is incident with n points, and each point is incident with r lines.
• Two points are incident with at most one common line.
• If the point p is not incident with the line l, then there is exactly one line l' incident with p which is disjoint from l.

We say that two lines are parallel if they are equal or disjoint. It follows from the last axiom that Euclid's parallel postulate holds: if two lines are both parallel to a third line, then they are parallel to one another.

Hence parallelism is an equivalence relation on the set of lines; each equivalence class contains n lines, covering the point set exactly.

A net of degree 2 is just a square grid. A net of degree 3 is equivalent to a Latin square. More generally, a net of order n and degree r is equivalent to a set of r-2 MOLS.

The dual of a net is a transversal design. Its nr points are partitioned into r groups each of size n; each block has size r, and is a transversal to the groups. Two points in different groups lie in a unique block.

Peter J. Cameron
2 August 2002