Monomial Flocks of Type II
Problem: A monomial flock is one whose coordinate functions
are all monomials, i.e., up to weak equivalence, the planes of the flock
have equations of the form taX + tbY + tcZ
+ W = 0 (t in GF(q)). A monomial cone is a cone with vertex (0,0,0,1)
whose point coordinates satisfy Yß
= XZß - 1, with (ß, q-1) = (ß - 1, q-1) = 1.
In almost all of the known examples where a monomial cone admits a monomial
flock, we have ß = (a-c)/(b-c) mod (q-1). These flocks are said to be
of Type I, and any other monomial flocks of monomial cones are said
to be of Type II. There are only two known examples of Type II flocks,
both in PG(3, 256). Are there other examples? (Most likely) Can they be
classified? (Seems difficult)
First Cited: Cherowitzo, W., "Monomial
Flocks of Monomial Cones in Even Characteristic".
- It can be shown that translation oval cones (this includes the
quadratic case), can only admit Type I monomial flocks.
- Computer results show that, in even characteristic, only Type I flocks
exist for q < 28, and for q = 29. There are no
other examples in PG(3,256) besides the two given in the paper.
Comments: The two examples are related to subfields of
GF(28), as is pointed out in the remarks following Theorem 3.1 of
the above cited paper.