Encyclopaedia of DesignTheory: Nets
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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.
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Peter J. Cameron
2 August 2002