  Motivated by applications in polymer-based data storage we introduced the new
problem of characterizing the code rate and designing constant-weight binary
$B_2$-sequences. Binary $B_2$-sequences are collections of binary strings of
length $n$ with the property that the real-valued sums of all distinct pairs of
strings are distinct. In addition to this defining property, constant-weight
binary $B_2$-sequences also satisfy the constraint that each string has a
fixed, relatively small weight $\omega$ that scales linearly with $n$. The
constant-weight constraint ensures low-cost synthesis and uniform processing of
the data readout via tandem mass spectrometers. Our main results include upper
bounds on the size of the codes formulated as entropy-optimization problems and
constructive lower bounds based on Sidon sequences.
