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.

翻译：受基于聚合物的数据存储应用启发，我们提出了一个新的问题：刻画编码率并设计常权二进制$B_2$-序列。二进制$B_2$-序列是长度为$n$的二进制字符串集合，其性质是所有不同字符串对的实数值和互不相同。除这一核心性质外，常权二进制$B_2$-序列还要求每个字符串具有固定的、相对较小的权$\omega$，且$\omega$随$n$线性增长。常权约束确保了低成本的合成以及通过串联质谱仪对数据读出进行均匀处理。我们的主要结果包括：以熵优化问题形式给出的码本大小的上界，以及基于Sidon序列的构造性下界。