SIAM Journal on Computing, Volume 50, Issue 2, Page 301-349, January 2021.
We strengthen the notion of double samplers, first introduced by Dinur and Kaufman [``High dimensional expanders imply agreement expanders,” in Proc. 58th IEEE Symp. on Foundations of Comp. Science, IEEE, 2017, pp. 974--985], which are samplers with additional combinatorial properties, and whose existence we prove using high-dimensional expanders. The ABNNR code construction [N. Alon et al., IEEE Trans. Inform. Theory, 38 (1992), pp. 509--516] achieves large distance by starting with a base code $C$ with moderate distance, and then amplifying the distance using a sampler. We show that if the sampler is part of a larger double sampler, then the construction has an efficient list-decoding algorithm. Our algorithm works even if the ABNNR construction is not applied to a base code $C$ but rather to any string. In this case the resulting code is approximate-list-decodable, i.e., the output list contains an approximation to the original input. Our list-decoding algorithm works as follows: It uses a local voting scheme from which it constructs a unique games constraint graph. The constraint graph is an expander, so we can solve unique games efficiently. These solutions are the output of the list-decoder. This is a novel use of a unique games algorithm as a subroutine in a decoding procedure, as opposed to the more common situation in which unique games are used for demonstrating hardness results. Double samplers and high-dimensional expanders are akin to pseudorandom objects in their utility, but they greatly exceed random objects in their combinatorial properties. We believe that these objects hold significant potential for coding theoretic constructions and view this work as demonstrating the power of double samplers in this context.

中文翻译：

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使用双采样器进行列表解码

SIAM Journal on Computing，第 50 卷，第 2 期，第 301-349 页，2021 年 1 月。
我们加强了双采样器的概念，首先由 Dinur 和 Kaufman 引入[``高维扩展器意味着协议扩展器，”在 Proc. 第 58 届 IEEE 症状。在 Comp 的基础上。Science, IEEE, 2017, pp. 974--985]，它们是具有附加组合属性的采样器，我们使用高维扩展器证明了它们的存在。ABNNR 代码构建 [N. Alon 等人，IEEE Trans。通知。Theory, 38 (1992), pp. 509--516] 通过从具有中等距离的基本代码 $C$ 开始，然后使用采样器放大距离来实现大距离。我们表明，如果采样器是更大的双采样器的一部分，那么该构造具有有效的列表解码算法。即使 ABNNR 构造不是应用于基本代码 $C$ 而是应用于任何字符串，我们的算法也能工作。在这种情况下，结果代码是近似列表可解码的，即输出列表包含原始输入的近似值。我们的列表解码算法的工作原理如下：它使用本地投票方案，从中构建唯一的游戏约束图。约束图是一个扩展器，因此我们可以有效地解决独特的游戏。这些解决方案是列表解码器的输出。这是在解码过程中将独特游戏算法用作子程序的一种新颖用途，这与使用独特游戏来演示硬度结果的更常见情况相反。双采样器和高维扩展器在其效用上类似于伪随机对象，但它们在组合特性上大大超过随机对象。