## 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 t^{a}X + t^{b}Y + t^{c}Z
+ W = 0 (t in GF(q)). A *monomial cone* is a cone with vertex (0,0,0,1)
whose point coordinates satisfy Y^{ß}
= XZ^{ß - 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:**

- 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 < 2
^{8}, and for q = 2^{9}. 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(2^{8}), as is pointed out in the remarks following Theorem 3.1 of
the above cited paper.

Open Problems
Index