Computing the mode of XOR subsequences

Question

I was confronted with this problem in an online programming challenge and it has been bugging me since:

In the problem, you are given a list of 16-bit numbers, say $a_0, a_1, ..., a_n$. An "XOR subsequence" is defined as the exclusive-or combination of every element in a consecutive subsequence of the list. The problem is that you must find the most commonly occurring XOR subsequence out of every possible consecutive subsequence in the list. In the event of a tie, the lowest numeric XOR value out of the most frequently occurring ones wins.

The first solution I came up with was an $O(n^2)$ solution where I compute the XOR for consecutive subsequences $a_0,...,a_i$ from $i=0$ to $i=n$, which can be done in $O(n)$, and doing the same thing starting at $a_1$, etc, and incrementing a counter in a map for every result. I know this repeats a lot of work, so I tackled the problem from a different angle and tried to define it mathematically. I defined $X_{i,j}$ to be the XOR of $a_i,...,a_j$ so that I could write the following:

$X_{i,j} = \begin{cases} a_i & \text{if } i = j \\ a_j \oplus X_{0, j-1} & \text{if } i = 0 \text{ and } j > 0 \\ X_{0,i-1} \oplus X_{0,j} & \text{if } 0 < i < j \end{cases} $

I implemented that with memoization, which I believe eliminates all redundant calculations, but using memoization turns out to be slow and use $O(n^2)$ space when attempting to calculate all $X_{i,j}$ for $0 \leq i \leq n $ and $i \leq j \leq n$, which is unacceptable.

Is there any way I can make this any faster than $O(n^2)$? Or if not, is there any efficient way to reduce the number of computations? Or am I missing some simplification to the problem that doesn't require me to compute every $X_{i,j}$? I'm sort of new to dynamic programming, so it's been hard to wrap my brain around this problem. My original solution was too slow by a factor of about 5 seconds (vs the time limit of 4 seconds for the challenge), so I believe even a constant factor reduction in time would make the solution acceptable.