有效树码及其应用：交互通信编码
Potent Tree Codes and their applications: Coding for Interactive Communication, revisited
摘要：
我们研究的基本问题，可靠的交互式通信在嘈杂的信道。在1992年和1993年发表的一系列突破性论文中，舒尔曼给出了非建设性的证明，证明了存在通用方法来模拟任何两方交互协议，例如：（1）仿真协议采用比原始协议长的常数因子，以及（2）如果仿真协议在噪声信道上执行，则仿真协议失败的概率在协议的总长度中呈指数地小。不幸的是，舒尔曼的仿真过程要么只在具有大量共享随机性的模型中工作，要么是非建设性的，因为它们依赖于良好的树代码的存在。唯一已知的证明存在良好的树码是非建设性的，并找到一个明确的建设仍然是一个重要的开放问题。事实上，随机生成的树码并不是具有压倒性概率的好树码。在这项工作中，我们重新审视的问题，可靠的交互式通信，并获得以下结果：我们引入了一个新的概念的善良的树代码，并定义了一个有效的树代码的概念。我们认为，这一概念具有独立的意义。我们证明了基于任何有效的树代码的显式仿真过程的正确性。我们表明，一个随机生成的树码（适当的恒定字母大小）是一个强大的树码具有压倒性的概率。此外，我们能够部分去随机化这个结果只使用O（n）随机位，其中$n$是树的深度。这些结果使我们能够获得第一个完全明确的仿真程序，可靠的交互式通信噪声信道与恒定的通信开销，和指数小的故障概率。
英文摘要：
We study the fundamental problem of reliable interactive communication over a noisy channel. In a breakthrough sequence of papers published in 1992 and 1993, Schulman gave non-constructive proofs of the existence of general methods to emulate any two-party interactive protocol such that: (1) the emulation protocol takes a constant-factor longer than the original protocol, and (2) if the emulation protocol is executed over a noisy channel, then the probability that the emulation protocol fails is exponentially small in the total length of the protocol. Unfortunately, Schulman's emulation procedures either only work in a model with a large amount of shared randomness, or are non-constructive in that they rely on the existence of good tree codes. The only known proofs of the existence of good tree codes are non-constructive, and finding an explicit construction remains an important open problem. Indeed, randomly generated tree codes are not good tree codes with overwhelming probability. In this work, we revisit the problem of reliable interactive communication, and obtain the following results: We introduce a new notion of goodness for a tree code, and define the notion of a potent tree code. We believe that this notion is of independent interest. We prove the correctness of an explicit emulation procedure based on any potent tree code. We show that a randomly generated tree code (with suitable constant alphabet size) is a potent tree code with overwhelming probability. Furthermore we are able to partially derandomize this result using only O(n) random bits, where $n$ is the depth of the tree. These results allow us to obtain the first fully explicit emulation procedure for reliable interactive communication over noisy channels with a constant communication overhead, and exponentially small failure probability.
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