[论文解读] List Decoding of Insertion and Deletion Codes

Shu Liu, Ivan Tjuawinata+1 more|arXiv (Cornell University)|Jun 24, 2019

DNA and Biological Computing参考文献 22被引用 3

一句话总结

本文在插入数超过删除数且字母表较大的情况下，证明了随机插入-删除码在超出Singleton界范围内的列表可译码性，证明了存在具有多项式列表大小的码，其纠错能力超过其最小插入-删除距离所允许的范围。此外，本文构造了具有高效列表译码的显式插入-删除码，并推导出插入-删除错误的Zyablov型界。

ABSTRACT

Insdel errors occur in communication systems caused by the loss of positional information of the message. Since the work by Guruswami and Wang, there have been some further investigations on the list decoding of insertion codes, deletion codes and insdel codes. However, unlike classical Hamming metric or even rank-metric, there are still many unsolved problems on list decoding of insdel codes. The contributions of this paper mainly consist of two parts. Firstly, we analyze the list decodability of random insdel codes. We show that list decoding of random insdel codes surpasses the Singleton bound when there are more insertion errors than deletion errors and the alphabet size is sufficiently large. Furthermore, our results reveal the existence of an insdel code that can be list decoded against insdel errors beyond its minimum insdel distance while still having polynomial list size. This provides a more complete picture on the list decodability of insdel codes when both insertion and deletion errors happen. Secondly, we construct a family of explicit insdel codes with efficient list decoding algorithm. As a result, we derive a Zyablov-type bound for insdel errors. Recently, after our results appeared, Guruswami et al. provided a complete solution for another open problem on list decoding of insdel codes. In contrast to the problems we considered, they provided a region containing all possible insertion and deletion errors that are still list decodable by some q-ary insdel codes of non-zero rate. More specifically, for a fixed number of insertion and deletion errors, while our paper focuses on maximizing the rate of a code that is list decodable against that amount of insertion and deletion errors, Guruswami et al. focuses on finding out the existence of a code with asymptotically non-zero rate which is list decodable against this amount of insertion and deletion errors.

研究动机与目标

• 研究在插入与删除错误不对称时，随机插入-删除码的列表可译码性。
• 确定插入-删除码是否能以多项式列表大小实现超越其最小插入-删除距离的列表译码。
• 构造具有高效列表译码算法的显式插入-删除码。
• 推导插入-删除错误校正码的Zyablov型界。
• 为Guruswami等人近期关于列表译码插入-删除错误存在非零速率码的研究提供互补视角。

提出的方法

• 使用概率方法分析在不对称错误模型下随机插入-删除码的列表可译码性。
• 证明当插入错误数超过删除错误数且字母表大小足够大时，列表译码半径可超越Singleton界。
• 利用代数-几何码与级联码构造方法，构建一族显式插入-删除码。
• 基于插值与求根技术，设计所构造码的高效列表译码算法。
• 通过优化构造中码率与最小插入-删除距离的权衡，推导出Zyablov型界。
• 与Guruswami等人近期工作进行比较，突出在插入-删除码列表译码领域中互补的贡献。

实验结果

研究问题

• RQ1 当插入错误数超过删除错误数且字母表较大时，随机插入-删除码是否能实现超越Singleton界的列表译码？
• RQ2 是否存在一种插入-删除码，其列表大小为多项式，且能纠正超过其最小插入-删除距离所允许的错误数？
• RQ3 能否构造出具有高效列表译码算法和非平凡码率的显式插入-删除码？
• RQ4 在插入与删除错误不对称的情况下，插入-删除码的列表译码可达率区域是什么？
• RQ5 与Guruswami等人近期提出的非零速率码存在性结果相比，本文的构造在码率与纠错容忍度方面有何差异？

主要发现

• 当插入错误数超过删除错误数且字母表大小足够大时，随机插入-删除码的列表译码可超越Singleton界。
• 存在一种插入-删除码，其列表可译码的插入-删除错误数超过其最小插入-删除距离所允许的范围，同时保持多项式列表大小。
• 构造了一族具有高效列表译码算法的显式插入-删除码，支持实际应用。
• 推导出插入-删除错误的Zyablov型界，为显式码构造提供了码率-距离权衡。
• 本研究结果与Guruswami等人近期工作形成互补，后者在更广泛的错误区域内建立了列表译码插入-删除错误非零速率码的存在性。
• 本文在不对称错误模式下，为插入-删除码的列表可译码性提供了更完整的图景。