Finding all polynomials up to a given degree to generate a bit sequence Consider the following bit sequence:

00100000110

Using Berlekamp-Massey algorithm we can find the following minimal degree polynomial:

$P(x)=x^6 + x^5 + x^4 + 1$

Now, is there a way to find all polynomials up to a given degree 8 that can generate the same bit sequence?
Given any finite sequence $(a_0,a_1,\ldots,a_{N-1})$ we may be interested in generating exactly that sequence, or generating the minimal degree (periodic) sequence which has that sequence as its' starting bits.

Trivially, the circulating shift register of length $L$, initially loaded by $(a_0,\ldots,a_{L-1})$, will generate this sequence, by implementing the recurrence

$$x_{t}=x_{t-L},\quad t\geq L,$$ 

i.e., give the characteristic polynomial $X^L+1.$

The polynomial that Berlekamp Massey (BM) yields is the minimal degree polynomial which matches this initial segment. Your sequence $$(a_0,\ldots,a_{10})=(00100000110)$$ has length $N=11$ and BM gave you a  degree $6$ polynomial.

In general, there will be a set of polynomials that can generate an initial segment. If the minimal recurrence (i.e., the minimal degree polynomial generating your sequence) has degree $n$, $2n$ bits are required to uniquely determine it. So to get a unique polynomial of degree 8 which generates a sequence, 16 bits are needed.

If you specify less than 16 bits, you will in general get more than one generating polynomial. You have 5 bits "free" to go from 11 to 16 bits so you may get up to 32 different polynomials, corresponding to all these possibilities, barring any linear dependencies in the resulting polynomials.