Let $x_1, \ldots,x_N \in \{0,1\}^D$ be $N$ binary vectors, assumed affinely independent (in the field of reals). Is there an efficient algorithm for determining whether a new binary vector $x_{N+1}$ is in the $\mathbb{R}$-affine span of $x_1,\ldots,x_N$?
By "efficient" I mean faster than having to test linear independence, i.e., something that exploits the fact that the vectors are binary.