We are given the sequence $b_i$ for $i$ multiple of $m>1$ with $0\le i<2n\,m$.
- If all the given $b_i$ are identical, we can't compute any missing $b_i$ (beyond a wild guess they are all $b_0$), stop.
- Build the sequence of $2n$ known bits $c_j=b_{(j\,m)}$ with $0\le j<2n$.
- Use Berlekamp–Massey to find a LFSR (polynomial $P$ of degree $k$ and $k$-bit initial state) matching $c_j$.
- Find some period $p$ of the $c_j$. For small degree $k$, we can simulate operation of the LFSR until the state repeats (OK to say $k\le40$ on a PC) or we can use baby-step/giant-step (OK to say $k\le75$ on a PC, see final section of this former revision). But more generally, if $P$ factors into a product of irreducible polynomials of degrees $k_r$ (thus with $k=\sum k_r$), then a period is $p=\prod(2^{k_r}-1)$.
Note: For large polynomial, we can still efficiently compute $c_j$ for large arbitrary $j$ (see this), once we have $P$ and starting state.
- If $\gcd(m,p)=1$, then $b_i$ also has period $p$, thus $\forall j\in\Bbb N,\ c_j=b_{(j\,m\bmod p)}$ ; it follows that $\forall i\in\Bbb N, b_i=c_{(i\,m^{-1}\bmod p)}$
Note: $m^{-1}\bmod p$ needs to be computed only once, e.g. with the half extended Euclidean algorithm.
Note: with $p=\prod(2^{k_r}-1)$ and $m$ a power of two as in the question, $\gcd(m,p)=1$ always holds. If $\gcd(m,p)\ne1$, it might be possible to find a smaller period, by factoring $p$ and iteratively removing those factors that leave what remains a period, until finding the smallest period $p_\text{min}$ (which depends on the irreducible factors of $P$ being primitive or not, and on the starting point). $\gcd(m,p_\text{min})=1$ allows to conclude without guesswork.
Once $2k$ consecutive $b_j$ are found, an easy option to find the LFSR yielding them is to use Berlekamp–Massey again. The initial state of the LFSR for the $b_j$ (in Fibonacci form) is of course the first $k$ bits $b_i$.
Not covered (yet?): polynomial arithmetic would allow to directly derive the LFSR for the $b_j$ from the one for the $c_j$; and conclude when $\gcd(m,p_\text{min})\ne1$ and partial information on the $b_j$ is available.
