Deletion-correcting codes for an adversarial nanopore channel

TL;DR

The paper addresses deleting errors in a nanopore channel modeled as an ISI system with window $\ell$ followed by an adversarial deletion channel that removes at most $t$ $\ell$-mers. It introduces an explicit $q$-ary code construction of length $n$ with redundancy $2t\log_q n+\Theta(\log\log n)$ by reducing deletion-correction to a near-Hamming problem using periodic consistency patterns and generalized Reed-Solomon codes to encode a check-pattern length vector, which is further protected by a $q$-ary $t$-deletion-correcting code. The authors derive existential bounds on the optimal redundancy, showing it lies between $t\log_q n+\Omega(1)$ and $2t\log_q n-\log_q\log_2 n+O(1)$, and the explicit construction matches the first-order term of these bounds. This work sharpens understanding of deletion-correcting capabilities in adversarial nanopore-like channels and provides near-optimal, explicit codes suitable for DNA data storage scenarios affected by deletions.

Abstract

We study deletion-correcting codes for an adversarial nanopore channel in which at most $t$ deletions may occur. We propose an explicit construction of $q$-ary codes of length $n$ for this channel with $2t\log_q n+Θ(\log\log n)$ redundant symbols. We also show that the optimal redundancy is between $t\log_q n+Ω(1)$ and $2t\log_q n-\log_q\log_2 n+O(1)$, so our explicit construction matches the existential upper bound to first order. In contrast, for the classical adversarial $q$-ary deletion channel, the smallest redundancy achieved by known explicit constructions that correct up to $t$ deletions is $4t(1+ε)\log_q n+o(\log n)$.

Theorems & Definitions (25)

• Lemma 1: xie2025two
• Proposition 2
• Definition 1: Periodic patterns
• Theorem 3: Fine-Wilf theorem fine1965uniqueness
• Remark 2.1
• Remark 2.2
• Lemma 4
• proof
• Definition 2: Consistency-preserving pattern
• Remark 2.3