发送带有未知噪音的消息
Sending a Message with Unknown Noise
摘要：
Alice和Bob通过双向信道连接，Alice想向Bob发送一条$L$位的消息。对手在信道上翻转任意但有限数量的比特$T$。这个对手知道我们的算法和Alice的消息，但不知道Alice或Bob生成的任何私有随机比特，也不知道通过信道发送的比特，除非这些比特可以通过Alice的消息或我们的算法的知识来预测。我们希望Bob收到Alice的消息，并且两个参与者都终止，错误概率最多为$δ> 0$，其中$δ$是Alice和Bob都知道的参数。不幸的是，值$T$对于Alice或Bob事先是未知的，并且值$L$对于Bob事先是未知的。我们描述了一种算法来解决上述问题，同时发送预期的$L + O \left( T + \min \left(T+1,\frac{L}{\log L} \right) \log \left( \frac{L}δ \right) \right)$位。一个特殊的情况是当$δ= O(1/L^c)$，对于一些恒定的$c$。然后，当$T = o(L/\log L)$时，发送的预期比特数是$L + o(L)$，当$T = Ω(L)$时，发送的预期比特数是$L + O\left( T \right)$，这是渐进最优的。
英文摘要：
Alice and Bob are connected via a two-way channel, and Alice wants to send a message of $L$ bits to Bob. An adversary flips an arbitrary but finite number of bits, $T$, on the channel. This adversary knows our algorithm and Alice's message, but does not know any private random bits generated by Alice or Bob, nor the bits sent over the channel, except when these bits can be predicted by knowledge of Alice's message or our algorithm. We want Bob to receive Alice's message and for both players to terminate, with error probability at most $δ> 0$, where $δ$ is a parameter known to both Alice and Bob. Unfortunately, the value $T$ is unknown in advance to either Alice or Bob, and the value $L$ is unknown in advance to Bob. We describe an algorithm to solve the above problem while sending an expected $L + O \left( T + \min \left(T+1,\frac{L}{\log L} \right) \log \left( \frac{L}δ \right) \right)$ bits. A special case is when $δ= O(1/L^c)$, for some constant $c$. Then when $T = o(L/\log L)$, the expected number of bits sent is $L + o(L)$, and when $T = Ω(L)$, the expected number of bits sent is $L + O\left( T \right)$, which is asymptotically optimal.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
