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New Construction of $q$-ary Codes Correcting a Burst of at most $t$ Deletions

Wentu Song, Kui Cai, Tony Q. S. Quek

TL;DR

This work tackles constructing $q$-ary burst-deletion codes that correct a burst of at most $t$ deletions with near-optimal redundancy and near-linear encoding/decoding. It introduces $q$-ary pattern-dense sequences and a VT-like locator, together with syndrome-compression and a sequence-replacement encoding framework (EncDen/DecDen), to achieve a redundancy of $\log n+8\log\log n+o(\log\log n)+\gamma_{q,t}$ bits and time $O(q^{7t} n (\log n)^3)$ for encoding/decoding. A key feature is that the second redundancy term is independent of $q$ and $t$, improving upon previous $\log n+O(\log q\log\log n)$ or $\log n+O(t^2\log\log n)$ bounds. The paper also discusses potential extensions to reconstruction codes with multiple reads, implying practical relevance for DNA storage and synchronization systems.

Abstract

In this paper, for any fixed positive integers $t$ and $q>2$, we construct $q$-ary codes correcting a burst of at most $t$ deletions with redundancy $\log n+8\log\log n+o(\log\log n)+γ_{q,t}$ bits and near-linear encoding/decoding complexity, where $n$ is the message length and $γ_{q,t}$ is a constant that only depends on $q$ and $t$. In previous works there are constructions of such codes with redundancy $\log n+O(\log q\log\log n)$ bits or $\log n+O(t^2\log\log n)+O(t\log q)$. The redundancy of our new construction is independent of $q$ and $t$ in the second term.

New Construction of $q$-ary Codes Correcting a Burst of at most $t$ Deletions

TL;DR

This work tackles constructing -ary burst-deletion codes that correct a burst of at most deletions with near-optimal redundancy and near-linear encoding/decoding. It introduces -ary pattern-dense sequences and a VT-like locator, together with syndrome-compression and a sequence-replacement encoding framework (EncDen/DecDen), to achieve a redundancy of bits and time for encoding/decoding. A key feature is that the second redundancy term is independent of and , improving upon previous or bounds. The paper also discusses potential extensions to reconstruction codes with multiple reads, implying practical relevance for DNA storage and synchronization systems.

Abstract

In this paper, for any fixed positive integers and , we construct -ary codes correcting a burst of at most deletions with redundancy bits and near-linear encoding/decoding complexity, where is the message length and is a constant that only depends on and . In previous works there are constructions of such codes with redundancy bits or . The redundancy of our new construction is independent of and in the second term.
Paper Structure (5 sections, 7 theorems, 46 equations, 2 figures)

This paper contains 5 sections, 7 theorems, 46 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pattern Dense Sequences
  4. Burst-deletion correcting $q$-ary codes
  5. Conclusions and Discussions

Key Result

Lemma 1

Tenengolts84 For any $\bm x\in\Sigma_q^n$, given $\mathsf{VT}(\phi(\bm x)_{[2,n]})$, $\mathsf{Sum}(\bm x)$ and any $\bm y\in\mathcal{D}_1(\bm x)$, one can uniquely recover $\bm x$, where $\phi(\bm x)_{[2,n]}=(\phi(\bm x)_2,\cdots,\phi(\bm x)_{n})$ and

Theorems & Definitions (16)

  • Lemma 1
  • Lemma 2
  • proof 1
  • Definition 1
  • Remark 1
  • Lemma 3
  • proof 2
  • Lemma 4
  • proof 3
  • Lemma 5
  • ...and 6 more