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

TL;DR

This paper establishes a fundamental trade-off in optimal reconstruction codes over channels with multiple burst substitutions, providing bounds, algorithms, and combinatorial insights for improved error correction and reconstruction.

Contribution

It introduces a new trade-off relation among error correction, reads, and list size, along with sharp bounds and an efficient list-reconstruction algorithm for burst-substitution channels.

Findings

Derived sharp asymptotic bounds on error ball sizes in burst metric.

Improved Gilbert-Varshamov bound for multiple bursts.

Proposed an efficient list-reconstruction algorithm.

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