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.
Motivation and more general question
$g = x^{16}+x^{12}+x^5 +1$ is generating polynom for the CRC-16-CCITT 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 ?
Examples
k =1,2 , answer = 0, rather obviously for all N.
k = 2 , answer = 0, (Wrong as Douglas Zare pointed in his answer)
k= 3 , answer = 0 , AFAIU (=as far as I understand)
k= 4 , answer N-15 , AFAIU (Wrong as Douglas Zare pointed in his answer)
Some guess based on numerical experiments
it seems the distribution is Gaussian like near its maximum - it seems that it does not depend much on polynomial (only tails depends), so we can take polynomial to be just g=x^16, for which the answer is obviusly binomial coefficient (N-16, k), which behaves like Gaussian by central limit theorem.
Questions Is this guess reasonable ? It is is correct what is the deviation of real distribution and gaussian ? What happens with tails ?