Codes for Correcting $t$ Limited-Magnitude Sticky Deletions

TL;DR

This paper introduces new coding schemes to correct $t$-sticky deletions with limited-magnitude errors, providing both non-systematic and systematic constructions with explicit redundancy bounds for applications in data storage systems.

Contribution

It proposes the first non-systematic and systematic codes specifically designed for $t$-sticky deletions with limited-magnitude errors, including efficient encoding and decoding algorithms.

Findings

Redundancy for non-systematic code: $2t(1-1/p) imes ext{log}(n+1) + O(1)$

Redundancy for systematic code: $rac{ ext{ceil}(2t(1-1/p)) imes ext{ceil}( ext{log} p)}{ ext{log} p} imes ext{log}(n+1) + O( ext{log} ext{log} n)$

Codes are suitable for flash memories, racetrack memories, and DNA data storage systems.

Abstract

Codes for correcting sticky insertions/deletions and limited-magnitude errors have attracted significant attention due to their applications of flash memories, racetrack memories, and DNA data storage systems. In this paper, we first consider the error type of $t$-sticky deletions with $\ell$-limited-magnitude and propose a non-systematic code for correcting this type of error with redundancy $2t(1-1/p)\cdot\log(n+1)+O(1)$, where $p$ is the smallest prime larger than $\ell+1$. Next, we present a systematic code construction with an efficient encoding and decoding algorithm with redundancy $\frac{\lceil2t(1-1/p)\rceil\cdot\lceil\log p\rceil}{\log p} \log(n+1)+O(\log\log n)$, where $p$ is the smallest prime larger than $\ell+1$.