With primitive polynomials, it's not too hard to get all the polynomials of a particular power. For example, columns in the following represent the 18, 16, 48, and 60 primitive polynomials of GF[2^7], GF[2^8], GF[2^9], and GF[2^10]:
You can also enter at the left, visit every white space, and exit at the right. It's a continuous connected cavern.
But here, I got lost in the caverns of GF[2^12]. I wasn't able to visit all the white spaces, and likely got eaten by a grue. Is there an optimal way to shuffle this set of primitive polynomial coefficients?
Here's more bad, dangerous caverns for all the primitive polynomials in caverns for powers 5 to 14.
Can the orderings of these primitive polynomials be made as safe as the solution for GF[2^7]-GF[2^10] at the top? Here are best found so for for GF[2^11] and GF[2^12]
A large bounty by Archmage Gruber prompted me to take another excursion into the caverns of the Galois Fields. It's said that some young fellow met his death in a duel amongst these grounds. Despite the obvious dangers, I delved in with new equipment. I wasn't able to improve GF[2^11] and GF[2^12] all that much, but at least I escaped unscathed at the other side. I should likely try giving up on symmetry, but it's hard to do that while traveling in a group.
I also ventured into GF[2^13] and GF[2^14]. It's difficult just to get to the other side with these caverns, but I spent a few days trying to make them safer. But I doubt that making them completely safe is possible.
Not the bounty of success that I had hoped for. But an interesting excursion, nevertheless.
UPDATE: GF[2^11] perfectly solved. Here are two solutions.
UPDATE 2: GF[2^12] almost perfectly solved.