Derivable affine planes and translation planes

Affine and projective planes

Mubs Inequivalence and Affine Planes

There are fairly large families of unitarily inequivalent complete sets of N+1 mutually unbiased bases (MUBs) in C for various prime powers N . The number of such sets is not bounded above by any polynomial as a function of N . While it is standard that there is a superficial similarity between complete sets of MUBs and finite affine planes, there is an intimate relationship between these large...

Convexity in Topological Affine Planes

We extend to topological affine planes the standard theorems of convexity, among them the separation theorem, the anti-exchange theorem, Radon’s, Helly’s, Carathéodory’s, and Kirchberger’s theorems, and the Minkowski theorem on extreme points. In a few cases the proofs are obtained by adapting proofs of the original results in the Euclidean plane; in others it is necessary to devise new proofs ...

Nearly flag-transitive affine planes

Spreads of orthogonal vector spaces are used to construct many translation planes of even order q, for odd m > 1, having a collineation with a (q − 1)-cycle on the line at infinity and on each of two affine lines.

Translation Planes and Derivation Sets

Using ideas from algebraic coding theory, a general notion of a derivation set for a pro-jective plane is introduced. Certain geometric codes are used to locate such sets. These codes also lead to upper bounds for the pranks of incidence matrices of translation planes in terms of the dimensions of the associated codes.