Optimal Reconstruction Codes with Given Reads in Multiple Burst-Substitutions Channels
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913895456702464 |
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| author | Yu, Wenjun Sun, Yubo Xu, Zixiang Ge, Gennian Schwartz, Moshe |
| author_facet | Yu, Wenjun Sun, Yubo Xu, Zixiang Ge, Gennian Schwartz, Moshe |
| contents | We study optimal reconstruction codes over the multiple-burst substitution channel. Our main contribution is establishing a trade-off between the error-correction capability of the code, the number of reads used in the reconstruction process, and the decoding list size. We show that over a channel that introduces at most $t$ bursts, we can use a length-$n$ code capable of correcting $ε$ errors, with $Θ(n^ρ)$ reads, and decoding with a list of size $O(n^λ)$, where $t-1=ε+ρ+λ$. In the process of proving this, we establish sharp asymptotic bounds on the size of error balls in the burst metric. More precisely, we prove a Johnson-type lower bound via Kahn's Theorem on large matchings in hypergraphs, and an upper bound via a novel variant of Kleitman's Theorem under the burst metric, which might be of independent interest.
Beyond this main trade-off, we derive several related results using a variety of combinatorial techniques. In particular, along with tools from recent advances in discrete geometry, we improve the classical Gilbert-Varshamov bound in the asymptotic regime for multiple bursts, and determine the minimum redundancy required for reconstruction codes with polynomially many reads. We also propose an efficient list-reconstruction algorithm that achieves the above guarantees, based on a majority-with-threshold decoding scheme. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_12924 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal Reconstruction Codes with Given Reads in Multiple Burst-Substitutions Channels Yu, Wenjun Sun, Yubo Xu, Zixiang Ge, Gennian Schwartz, Moshe Information Theory Combinatorics We study optimal reconstruction codes over the multiple-burst substitution channel. Our main contribution is establishing a trade-off between the error-correction capability of the code, the number of reads used in the reconstruction process, and the decoding list size. We show that over a channel that introduces at most $t$ bursts, we can use a length-$n$ code capable of correcting $ε$ errors, with $Θ(n^ρ)$ reads, and decoding with a list of size $O(n^λ)$, where $t-1=ε+ρ+λ$. In the process of proving this, we establish sharp asymptotic bounds on the size of error balls in the burst metric. More precisely, we prove a Johnson-type lower bound via Kahn's Theorem on large matchings in hypergraphs, and an upper bound via a novel variant of Kleitman's Theorem under the burst metric, which might be of independent interest. Beyond this main trade-off, we derive several related results using a variety of combinatorial techniques. In particular, along with tools from recent advances in discrete geometry, we improve the classical Gilbert-Varshamov bound in the asymptotic regime for multiple bursts, and determine the minimum redundancy required for reconstruction codes with polynomially many reads. We also propose an efficient list-reconstruction algorithm that achieves the above guarantees, based on a majority-with-threshold decoding scheme. |
| title | Optimal Reconstruction Codes with Given Reads in Multiple Burst-Substitutions Channels |
| topic | Information Theory Combinatorics |
| url | https://arxiv.org/abs/2506.12924 |