I'm trying to construct a binary sequence of length $2^n$. This sequence will be converted to a square signal of $\pm 1$, where 0 produces $-1$ and 1 produces $1$. I want the resultant signal to be as spectrally flat as possible, minimizing the $L^2$ norm of the continuous Fourier transform. Is there a reason to believe that this problem is hard, such as by equivalence to a known mathematical problem like subset sum, or is there a solution I'm overlooking?
Things I've checked are Raskar's sequence, which uses exhaustive search, URA/MURA, and Levin 2007. The best method I've considered is genetic optimization, but I'd prefer an optimal solution over a searched one.
The motivation is that I have a reverb with Dirac delta spikes in pre-defined locations, and must choose a sign $\pm 1$ for each spike. Setting all signs to $-1$ produces a pitch determined by the locations, and setting signs randomly eliminates the pitch. This doesn't quite match the problem statement given above, but I think a solution for the above will lead to a solution for this.