Time/Space Optimal k-Subset Operator Application - Is this a named problem?
For $K \ll 2N/3$, a good solution goes as follows, for $K$ even (the case $K$ odd is very similar):


Generate all XORs of length 2.
Generate all XORs of length 3.
...
Generate all XORs of length $K/2$.
Generate all XORs of length $K$.


This requires $\binom{N}{K/2} + \binom{N}{K/2-1} \approx \binom{N}{K/2}$ memory and $\binom{N}{2} + \cdots + \binom{N}{K/2} + \binom{N}{K} \approx \binom{N}{K}$ operations. This is close to optimal in terms of number of operations, since clearly at least $\binom{N}{K}$ are needed.

You can optimize this a bit as follows:


Generate all XORs of length 2 of $X_1,\ldots,X_{N-K/2+2}$.
Generate all XORs of length 3 of $X_1,\ldots,X_{N-K/2+3}$.
...
Generate all XORs of length $K/2$ of $X_1,\ldots,X_N$.
Generate all XORs of length $K$.


The savings are not significant.