Optimal Reconstruction Codes with Given Reads in Multiple Burst-Substitutions Channels

2025-06-15

AI Summary

This paper investigates optimal code design for list reconstruction over multi-burst erasure channels, aiming to precisely characterize the trade-off among error-correction capability, read complexity, and list size. Methodologically, it establishes the first tight error-ball bound under the burst metric; derives a Johnson-type lower bound via Kahn’s theorem and a novel upper bound based on a variant of Kleitman’s theorem; and improves the Gilbert–Varshamov bound for multi-burst settings. Leveraging combinatorial design, hypergraph matching, and discrete geometric techniques, the authors construct an efficient list-reconstruction algorithm with provable guarantees: achieving $O(n^lambda)$-list decoding with $Theta(n^ ho)$ reads, subject to $t-1 = varepsilon + ho + lambda$, and determining the minimum redundancy achievable under polynomial read complexity. The results provide both tight theoretical bounds and constructive schemes for high-fault-tolerance, low-overhead data recovery.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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 $epsilon$ errors, with $Theta(n^ ho)$ reads, and decoding with a list of size $O(n^lambda)$, where $t-1=epsilon+ ho+lambda$. 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.

Problem

• Study optimal reconstruction codes for multiple-burst substitution channels
• Trade-off between error-correction, reads, and decoding list size
• Improve Gilbert-Varshamov bound for multiple bursts using combinatorial techniques

Innovation

• Trade-off between error-correction and reads
• Johnson-type lower bound via Kahn's Theorem
• Efficient list-reconstruction algorithm with threshold