Efficient Linear and Affine Codes for Correcting Insertions/Deletions
Efficient Linear and Affine Codes for Correcting Insertions/Deletions
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DOI：
10.1137/1.9781611976465.1
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发表时间：
2020-07
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影响因子：
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通讯作者：
Kuan Cheng;V. Guruswami;Bernhard Haeupler;Xin Li
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作者：
Kuan Cheng;V. Guruswami;Bernhard Haeupler;Xin Li
研究了用于纠正插入和删除等同步错误的EMPH{LINEAR}和\EMPH{AFINE}纠错码。我们称这种码为线性/仿射内码。甚至可以纠正单个删除的线性码被限制为具有至多$1/2$的信息率(通过平凡的2重重复码来实现)。以前，(错误地)报告说，更普遍地不存在纠正$k$删除的非平凡线性码，即，$(k+1)$-折重复码及其比率$1/(k+1)$对于任何$k$基本上是最优的。我们证明了这一点，并证明了长度为$n$、码率略低于$1/2$的二元线性码的存在能够纠正$\Omega(N)$的插入和删除。这将速率$1/2$确定为从线性码的删除中恢复的尖锐阈值，并重新开启了对更好地理解线性码用于纠正插入/删除的能力的探索。我们证明了线性INSDEL码的码率与(编辑)距离权衡的新的外界和存在内界。我们用一种有效的基于同步字符串的变换来补充我们的存在结果，该变换将任何用于Hamming误差的渐近良好的线性代码转换为用于Indel误差的渐近良好的线性代码。最后，通过给出一个速率为$1-\epsilon$的显式仿射编码，证明了$\FRAC{1}{2}$-速率限制对仿射编码不成立，它可以有效地纠正一定比例的INSTELL错误。
This paper studies \emph{linear} and \emph{affine} error-correcting codes for correcting synchronization errors such as insertions and deletions. We call such codes linear/affine insdel codes. Linear codes that can correct even a single deletion are limited to have information rate at most $1/2$ (achieved by the trivial 2-fold repetition code). Previously, it was (erroneously) reported that more generally no non-trivial linear codes correcting $k$ deletions exist, i.e., that the $(k+1)$-fold repetition codes and its rate of $1/(k+1)$ are basically optimal for any $k$. We disprove this and show the existence of binary linear codes of length $n$ and rate just below $1/2$ capable of correcting $\Omega(n)$ insertions and deletions. This identifies rate $1/2$ as a sharp threshold for recovery from deletions for linear codes, and reopens the quest for a better understanding of the capabilities of linear codes for correcting insertions/deletions. We prove novel outer bounds and existential inner bounds for the rate vs. (edit) distance trade-off of linear insdel codes. We complement our existential results with an efficient synchronization-string-based transformation that converts any asymptotically-good linear code for Hamming errors into an asymptotically-good linear code for insdel errors. Lastly, we show that the $\frac{1}{2}$-rate limitation does not hold for affine codes by giving an explicit affine code of rate $1-\epsilon$ which can efficiently correct a constant fraction of insdel errors.