关于Cohen-Haeupler-Schulman树码的译码
On Decoding Cohen-Haeupler-Schulman Tree Codes
摘要：
树码是由Schulman提出的一种组合结构，是交互式通信编码所必需的。速率和距离都有界的树码称为渐近好码。速率恒定等同于字母表大小恒定。舒尔曼证明，有渐近良好的树代码家庭使用洛瓦兹本地引理，但他们的明确建设仍然是一个突出的开放问题。在一项重大突破中，Cohen、Haeupler和Schulman构造了显式树码族，它们具有恒定的距离，但长度超过一个字母表的多对数。我们的主要结果是一个随机的多项式时间解码算法，这些代码使新的使用多项式的方法。校正的错误数量大致与块长度的四分之三次方成比例，达不到由恒定距离保证的恒定分数错误校正。我们进一步提出了一些理论的Cohen-Haeupler-Schulman代码的变体，所有纠正一个常数分数的错误与polylogarithmic字母表的大小。为了有效地纠正接近恒定比例的错误，我们提出了一种受压缩感知启发的投机性凸优化方法。
英文摘要：
Tree codes, introduced by Schulman, are combinatorial structures essential to coding for interactive communication. An infinite family of tree codes with both rate and distance bounded by positive constants is called asymptotically good. Rate being constant is equivalent to the alphabet size being constant. Schulman proved that there are asymptotically good tree code families using the Lovasz local lemma, yet their explicit construction remains an outstanding open problem. In a major breakthrough, Cohen, Haeupler and Schulman constructed explicit tree code families with constant distance, but over an alphabet polylogarithmic in the length. Our main result is a randomized polynomial time decoding algorithm for these codes making novel use of the polynomial method. The number of errors corrected scales roughly as the block length to the three-fourths power, falling short of the constant fraction error correction guaranteed by the constant distance. We further present number theoretic variants of Cohen-Haeupler-Schulman codes, all correcting a constant fraction of errors with polylogarithmic alphabet size. Towards efficiently correcting close to a constant fraction of errors, we propose a speculative convex optimization approach inspired by compressed sensing.
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