The Caverns of Primitive Polynomial GF[2]

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.