AI 中文总结
针对DNA频率编码中采样估算与测序噪声的问题，本文提出低复杂度编码方案，无条件实现$\boldsymbol{\text{log}_4 R}$比特传输率，并推广至噪声场景，以少量速率损失换取实用复杂度。
AI 中文摘要
DNA分子体积极小，因此利用其频率向量编码信息具备可行性。更具体地说，发送方可向溶液池中注入$M_X$份字符串$X=$CATCATCAT，接收方则可通过对溶液池测序来恢复$M_X$的数值。但该过程存在两类不确定性来源：（a）$M_X$通常数值过大，无法精确计数，只能通过采样估算；（b）DNA测序仪可能存在噪声，难以区分CATCATCAT与CATGATCAT这类序列。\n近期，Tamir、Weinberger与Guillén i Fàbregas明确了在（a）类不确定性下，频率向量可承载的信息量。他们证明每条字符串可承载约$\boldsymbol{\text{log}_4 R}$比特信息，其中$R$是每条字符串的平均读取次数；同时他们还指出，当不同字符串的数量至少为$\boldsymbol{\text{sqrt} R}$时，低复杂度的未编码方案即可实现$\text{log}_4 R$比特的信息传输率。本文中，我们证明一种低复杂度的编码方案可无条件实现相同的$\text{log}_4 R$比特传输率。随后我们将该方案推广至可应对（b）类测序噪声的场景，并证明噪声会使总比特数产生$\boldsymbol{\text{log}_2 \text{det} W}$的损失，此外由于证明过程中使用了傅里叶变换，还会带来一个线性项的损失。其中$\text{log}_2 \text{det} W$这一损失项与Gerzon、Shomorony和Weinberger得到的结果渐近一致；我们的方案以少量速率损失换取了实用的低复杂度。
英文摘要
DNA molecules are so small that it might be practical to use their frequency vectors to encode messages. More precisely, a sender can inject $M_X$ copies of the string $X =$ CATCATCAT into a pool and the receiver can recover $M_X$ by sequencing the pool. There are, however, two sources of uncertainty: (a) $M_X$ is usually too big to be counted exactly, but is estimated by sampling. (b) The DNA sequencer could be noisy; it may have difficulty distinguishing CATCATCAT from CATGATCAT. Recently, Tamir, Weinberger, and Guillén i Fàbregas clarified the amount of information the frequency vector can carry under (a). They showed that each string can carry about $\log_4 R$ bits, where $R$ is the average number of times each string is read. They also showed that $\log_4 R$ bits can be achieved by a low-complexity uncoded scheme under the condition that there are at least $\sqrt R$ distinct strings. In this paper, we show that a low-complexity coded scheme can achieve the same $\log_4 R$ bits unconditionally. We then generalize the scheme to handle sequencing noise, (b), and show that the noise penalizes the total number of bits by $\log_2 \det W$, together with a linear term due to the use of Fourier transforms in our proof. The former penalty $\log_2 \det W$ is asymptotically the same as that obtained by Gerzon, Shomorony, and Weinberger; our scheme trades a small amount of rate for practical complexity.
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