Unusual Sorting

You are given a sequence of distinct non-negative integers $A_1, A_2, \dots, A_N$. For every non-negative integer $x \lt 2^K$, let's define a function $f(x)$ as the number of inversions in the sequence $A_1 \oplus x, A_2 \oplus x, \dots, A_N \oplus x$. (An inversion in a sequence $X_1, X_2, \dots, X_N$ is a pair of indices $(i, j)$ such that $i \lt j$ and $X_i \gt X_j$.) Consider the sorted sequ