$t$-out-of-$n$ Secret Sharing over $\Bbb Z_2$? Is it possible to construct a $t$-out-of-$n$ secret sharing scheme over $\Bbb Z_2$?


Shamir Secret Sharing allows for an arbitrary threshold $t$ and an arbitrary number of participants $n$, but requires the field to have at least $n$ elements. Is there a similar construction that works over $\Bbb Z_2$?



Addendum

One approach could be to use Shamir SS over the finite field $\Bbb F_{2^{\log n}}$. Then, to share $x\in\Bbb Z_2$, this value is first embedded in this field and then it can be secret shared. However, this has an (apparently) innecessary overhead (even though it could be amortized if we share more than value)
The question does not ask specifically for being in Shamir. This is a very interesting question, and has been studied in depth, with Ronald Cramer leading the effort. I refer you to the paper Algebraic Geometric Secret Sharing Schemes and Secure Multi-Party Computations over Small Fields as a starting point in your research.