TITLE: The Caverns of Primitive Polynomial GF[2]
QUESTION [26 upvotes]: 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]

REPLY [5 votes]: 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.