Sorry, this is abit embarassing, but how exactly can I correct and update the descriptions?
I thought I find it easily when I was propsing my help, but it turns out, I do not find documentation on that or I am blind...

I enjoyed this kata a lot.
But there are two things that seem to be incorrect in the description:

1. Only matters, if you use the indeterminant exponent in anyway (i used it as an index):
   According to the explanations in the details, the generator-polynomial stays unchanged (always). Actually it needs to be adjusted after each of the up to nine calculation cycles.
   According to the linked step-by-step description, the coefficients of the generator-polynomial stay the same, but the indeterminant exponents need to be adjusted to fit the exponents of the message-polynomial.

Example:

Message polynomial after the first calculation cycle:
168 x^24 + 203 x^23 + 233 x^22 + ...
accordingly the generator polynomial's indeterminant exponents must be reduced by one:
a^0 x^24 + a^43 x^23 + a^139 x^22 + ...

In case two leading terms should be zero, the indeterminant exponents must be reduced by two.

2. was giving me errors on the second test string:
   In the description is a hint, that you need to modulo the exponents if they get greater than 255. Actually it must be done for exponents greater or equal to 255.
   alphaTable.exponentOf(255) = -1

I can correct that if you want me to.

Thanx for the detailed instructions otherwise, they made the kata accessible and fun.

According to the explanations in the details, the generator-polynomial stays unchanged (always). Actually it needs to be adjusted after each of the up to nine calculation cycles.
According to the linked step-by-step description, the coefficients of the generator-polynomial stay the same, but the indeterminant exponents need to be adjusted to fit the exponents of the message-polynomial.

Example:

  Message polynomial after the first calculation cycle:
  168 x^24 + 203 x^23 + 233 x^22 + ...
  accordingly the generator polynomial's indeterminant exponents must be reduced by one:
  a^0 x^24 + a^43 x^23 + a^139 x^22 + ...

In case two leading terms should be zero, the indeterminant exponents must be reduced by two.

This is only important if for some reason you use the exponents. In my case I used them as keys for the coefficients in a Map.