Deletion-correcting codes for an adversarial nanopore channel

TL;DR

This paper introduces an explicit construction of deletion-correcting codes for an adversarial nanopore channel, achieving near-optimal redundancy with a significant reduction compared to classical deletion channels.

Contribution

The paper presents a new explicit coding scheme for adversarial nanopore channels that nearly matches the theoretical lower bounds on redundancy.

Findings

Redundant symbols are $2t ext{log}_q n+ ext{Theta}( ext{log} ext{log} n)$ for the proposed codes.

The optimal redundancy lies between $t ext{log}_q n+ ext{Omega}(1)$ and $2t ext{log}_q n- ext{log}_q ext{log}_2 n+O(1)$.

Explicit codes outperform classical constructions in terms of redundancy for the same deletion correction capability.

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 $2t\log_q n+\Theta(\log\log n)$ redundant symbols. We also show that the optimal redundancy is between $t\log_q n+\Omega(1)$ 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 $4t(1+\epsilon)\log_q n+o(\log n)$.