Maximum XOR of Subsequences

Given an array, pick two (possibly overlapping or empty) subsequences and maximize the XOR of their XOR-sums — equivalently, choose two values from the set of all subset XORs and maximize their XOR. The core challenge is to characterize all achievable XORs and compute the maximum pairwise XOR efficiently (n up to 1e5, values up to 1e9), typically via a linear basis / Gaussian elimination in GF(2).