Let us consider polynoms over $F_2$.
Consider the linear SUBSPACE of polynoms divisible by   $x^{16}+x^{12}+x^5 +1$ and of degree less or equal $N$ (e.g. 40).

**Question**: How many k-nomials  belong to this subspace ? 

By k-nomials I mean polynom containing only $k$ monomials, e.g. x^2+x - is 2-nomial.

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Motivation and more general question

$g = x^{16}+x^{12}+x^5 +1$ is generating polynom for the [CRC-16-CCITT][1] error-correcting code. I am intersting about the Hamming weight distribution for the code-words,
it is important characteristics of the code.

**Question** More generally we can take other "generating polynoms" and ask a similar questions,
what is known about it ?

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Examples

k =1,2 , answer = 0,  rather obviously for all N.

k= 3 , answer = 0 , AFAIU (=as far as I understand)

k= 4 , answer N-15 , AFAIU 





  [1]: http://en.wikipedia.org/wiki/Cyclic_redundancy_check#Commonly_used_and_standardized_CRCs