Rings of geometries II

David G. Glynn

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)


Linear spaces are investigated using the general theory of "Rings of Geometries I." By defining geometries and ring structures in several different ways, formulae for linear spaces embedded in finite projective and affine planes are obtained. Several "fundamental theorems" of counting in finite projective planes are proved which show why configurations with at least three points per line and at least three lines through every point are important. These theorems are illustrated by finding the formulae for the number of k-arcs in a projective plane of order q for all k ≤ 8 and also by finding a formula for the number of blocking sets. A quick proof that a projective plane of order 6 does not exist follows from the formula for the number of 7-arcs in such a plane.

Original languageEnglish
Pages (from-to)26-66
Number of pages41
JournalJournal of Combinatorial Theory, Series A
Issue number1
Publication statusPublished - Sept 1988
Externally publishedYes


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