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".

Activity: Inactive.

Partial Results:

  1. It can be shown that translation oval cones (this includes the quadratic case), can only admit Type I monomial flocks.
  2. 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.

Open Problems