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Exact Reconstruction From Insertions in Synchronization Codes

机译:通过同步代码中的插入进行精确重构

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This paper studies problems in data reconstruction, an important area with numerous applications. In particular, we examine the reconstruction of binary and nonbinary sequences from synchronization (insertion/deletion-correcting) codes. These sequences have been corrupted by a fixed number of symbol insertions (larger than the minimum edit distance of the code), yielding a number of distinct traces to be used for reconstruction. We wish to know the minimum number of traces needed for exact reconstruction. This is a general version of a problem tackled by Levenshtein for uncoded sequences. We introduce an exact formula for the maximum number of common supersequences shared by sequences at a certain edit distance, yielding an upper bound on the number of distinct traces necessary to guarantee exact reconstruction. Without specific knowledge of the code words, this upper bound is tight. We apply our results to the famous single deletion/insertion-correcting Varshamov-Tenengolts (VT) codes and show that a significant number of VT code word pairs achieve the worst case number of outputs needed for exact reconstruction. We also consider extensions to other channels, such as adversarial deletion and insertion/deletion channels and probabilistic channels.
机译:本文研究数据重建中的问题,这是具有众多应用程序的重要领域。特别是,我们从同步(插入/删除校正)代码中检查了二进制和非二进制序列的重构。这些序列已被固定数量的符号插入(大于代码的最小编辑距离)破坏,从而产生了许多用于重构的不同迹线。我们希望知道精确重建所需的最少迹线数量。这是Levenshtein针对未编码序列解决的问题的一般版本。我们介绍了在一定的编辑距离处序列共享的最大公共超序列的最大数量的精确公式,从而为保证精确重构所必需的不同迹线的数量产生了上限。没有代码字的专门知识,这个上限是紧密的。我们将我们的结果应用于著名的单次删除/插入校正Varshamov-Tenengolts(VT)码,并表明大量VT码字对实现了精确重构所需的最坏情况下的输出。我们还考虑扩展到其他渠道,例如对抗性删除,插入/删除渠道和概率渠道。

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