• every line has n points;
• parallelism is an equivalence relation on lines, where two lines are parallel if they are disjoint or identical;
• there are k parallel classes each consisting of n lines;
• any two nonparallel lines meet exactly once.

Eric Moorhouse made a very interesting conjecture:

Let N_k be any k-net of order n, and let N_{k-1} be any (k-1)-subnet thereof. If p is a prime sharply dividing n then dim C_p(N_k) - dim C_p(N_{k-1}) is greater than or equal to n-k+1. (C_p(N_k) denotes the space generated by the incidence vectors of lines over the field with p elements.)

This conjecture was made in "Bruck Nets, Codes and Characters of Loops", which appeared in Designs, Codes and Cryptography, Vol. 1, 1991, pp. 7-29.

If this conjecture is true it would imply that all projective planes of square free order are desarguesian and that there are no planes of order congruent to 2 mod 4 except for n=2.

The interested reader should also check out Rene Peeters thesis "Ranks and Structure of Graphs" and his paper "On the p-ranks of Net Graphs" which appeared in Designs, Codes and Cryptography, vol. 5, 1995, 139-153.

You might also check out my papers:

• Nets and their codes. -- Designs, Codes and Cryptography, 3, 315-331, 1993.
  
• A Coding-Theoretic Solution to the 36 Officer Problem -- Designs, Codes and Cryptography, 4, 123-128, 1994.
  
• Planes, Nets and Codes -- Mathematical Journal of Okayama University Vol 38, December 1996.

Return to my home page.