For a very interesting paper that explains the relationship of codes and planes , see the paper by E.F. Assmus, Jr. and J.D. Key, "Affine and Projective Planes" which can be obtained from Assmus's home page.

I have conjectured that the Hull of a desarguesian affine plane of even order is determined by differences of parallel classes in the first n/2 parallel classes in any ordering of the parallel classes. I believe this is related to the hyperovals in the dual palne of the projective completion.

The conjecture is true for n=2,4,8, and 16. For more information see my paper: Planes, Nets and Codes -- Mathematical Journal of Okayama University Vol 38, December 1996.

Additional relationships can be found in Self-Dual Codes and Finite Projective Planes , Mathematical Journal of Okayama University, Vol. 40, 1998,(2000).

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