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Balanced reconstruction codes for single edits

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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.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, China

    Rongsheng Wu

  2. CAS Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, People’s Republic of China

    Xiande Zhang

  3. Hefei National Laboratory, University of Science and Technology of China, Hefei, 230088, China

    Xiande Zhang

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  1. Rongsheng Wu
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Correspondence to Xiande Zhang.

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The research of X. Zhang was supported by the National Key Research and Development Program of China 2020YFA0713100 and 2023YFA1010200, the NSFC under Grants Nos. 12171452 and 12231014, and the Innovation Program for Quantum Science and Technology 2021ZD0302902. The research of R. Wu was supported by China Postdoctoral Science Foundation under Grant No. 2021M703098.

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Wu, R., Zhang, X. Balanced reconstruction codes for single edits. Des. Codes Cryptogr. (2024). https://doi.org/10.1007/s10623-024-01377-y

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  • DOI: https://doi.org/10.1007/s10623-024-01377-y

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