Abstract

Motivated by the sequence reconstruction problem initiated by Levenshtein, reconstruction codes were introduced by Cai et al. to combat errors when a fixed number of noisy channels are available. The central problem on this topic is to design codes with sizes as large as possible, such that every codeword can be uniquely reconstructed from any N distinct noisy reads, where N is fixed. In this paper, we study binary reconstruction codes with the constraint that every codeword is balanced, which is a fundamental requirement in the technique of DNA-based storage. For all possible channels with a single edit error and their variants, we design asymptotically optimal balanced reconstruction codes for all N, and show that the number of their redundant symbols decreases from \(\frac{3}{2}\log _2 n+O(1)\) to \(\frac{1}{2}\log _2n+\log _2\log _2n+O(1)\) , and finally to \(\frac{1}{2}\log _2n+O(1)\) but with different speeds, where n is the length of the code. Compared with the unbalanced case, our results imply that the balanced property does not reduce the rate of the reconstruction code in the corresponding codebook.

中文翻译：

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单次编辑的平衡重建代码

摘要

受 Levenshtein 发起的序列重构问题的推动，Cai 等人引入了重构代码。当固定数量的噪声通道可用时，以防止错误。这个主题的中心问题是设计尽可能大的代码，使得每个代码字都可以从任何N个不同的噪声读取中唯一地重建，其中N是固定的。在本文中，我们研究了二进制重构码，其约束条件是每个码字都是平衡的，这是基于 DNA 的存储技术的基本要求。对于具有单个编辑错误的所有可能通道及其变体，我们为所有N设计渐近最优平衡重建码，并表明其冗余符号的数量从\(\frac{3}{2}\log _2 n+ O(1)\)到\(\frac{1}{2}\log _2n+\log _2\log _2n+O(1)\)，最后到\(\frac{1}{2}\log _2n +O(1)\)但速度不同，其中n是代码的长度。与不平衡情况相比，我们的结果表明平衡特性不会降低相应码本中的重构码的速率。