Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotically Optimal Codes for $(t,s)$-Burst Error

Yubo Sun, Ziyang Lu, Yiwei Zhang, Gennian Ge

TL;DR

This work addresses the design of efficient codes that can correct a $(t,s)$-burst error, which deletes $t$ consecutive symbols and inserts $s$ arbitrary symbols at the same position, a model motivated by DNA storage. The authors develop explicit $q$-ary constructions achieving redundancy $\log n + O(1)$ for any fixed $t,s$ and alphabet size $q$, by reducing general $(t,s)$-bursts to simpler canonical forms (notably $(2,2)$ and $(t',t'-1)$) via array representations and lifting techniques. They introduce novel sum-constraint VT-type codes for $(2,2)$-bursts, extend to binary and non-binary $(t,t-1)$-bursts, and then obtain $(t,s)$-burst codes with minimal additive overhead, along with several byproducts such as permutation codes for burst-stable deletions. The results close the gap to the sphere-packing bound up to an additive constant and offer broad applicability to related error types, including inversions and absorptions, with practical redundancy improvements for DNA storage and other burst-prone channels.

Abstract

Recently, codes for correcting a burst of errors have attracted significant attention. One of the most important reasons is that bursts of errors occur in certain emerging techniques, such as DNA storage. In this paper, we investigate a type of error, called a $(t,s)$-burst, which deletes $t$ consecutive symbols and inserts $s$ arbitrary symbols at the same coordinate. Note that a $(t,s)$-burst error can be seen as a generalization of a burst of insertions ($t=0$), a burst of deletions ($s=0$), and a burst of substitutions ($t=s$). Our main contribution is to give explicit constructions of $q$-ary $(t,s)$-burst correcting codes with $\log n + O(1)$ bits of redundancy for any given constant non-negative integers $t$, $s$, and $q \geq 2$. These codes have optimal redundancy up to an additive constant. Furthermore, we apply our $(t,s)$-burst correcting codes to combat other various types of errors and improve the corresponding results. In particular, one of our byproducts is a permutation code capable of correcting a burst of $t$ stable deletions with $\log n + O(1)$ bits of redundancy, which is optimal up to an additive constant.

Asymptotically Optimal Codes for $(t,s)$-Burst Error

TL;DR

This work addresses the design of efficient codes that can correct a -burst error, which deletes consecutive symbols and inserts arbitrary symbols at the same position, a model motivated by DNA storage. The authors develop explicit -ary constructions achieving redundancy for any fixed and alphabet size , by reducing general -bursts to simpler canonical forms (notably and ) via array representations and lifting techniques. They introduce novel sum-constraint VT-type codes for -bursts, extend to binary and non-binary -bursts, and then obtain -burst codes with minimal additive overhead, along with several byproducts such as permutation codes for burst-stable deletions. The results close the gap to the sphere-packing bound up to an additive constant and offer broad applicability to related error types, including inversions and absorptions, with practical redundancy improvements for DNA storage and other burst-prone channels.

Abstract

Recently, codes for correcting a burst of errors have attracted significant attention. One of the most important reasons is that bursts of errors occur in certain emerging techniques, such as DNA storage. In this paper, we investigate a type of error, called a -burst, which deletes consecutive symbols and inserts arbitrary symbols at the same coordinate. Note that a -burst error can be seen as a generalization of a burst of insertions (), a burst of deletions (), and a burst of substitutions (). Our main contribution is to give explicit constructions of -ary -burst correcting codes with bits of redundancy for any given constant non-negative integers , , and . These codes have optimal redundancy up to an additive constant. Furthermore, we apply our -burst correcting codes to combat other various types of errors and improve the corresponding results. In particular, one of our byproducts is a permutation code capable of correcting a burst of stable deletions with bits of redundancy, which is optimal up to an additive constant.
Paper Structure (24 sections, 29 theorems, 20 equations, 2 figures, 2 tables)

This paper contains 24 sections, 29 theorems, 20 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sphere Packing Bound on the Code Redundancy
  4. Connections Between Several Types of Burst Errors
  5. $(t,t)$-Burst and $(2,2)$-Burst where $t \geq 2$
  6. $(t,s)$-Burst and $(t',t'-1)$-Burst where $t>s$ and $t'= \lceil t/(t-s) \rceil+1$
  7. $(t,t)$-Burst Correcting Codes
  8. $(t,s)$-Burst Correcting Codes
  9. Binary $(t,t-1)$-Burst Correcting Codes
  10. The First Constraint
  11. The Second Constraint
  12. The Third Constraint
  13. Non-Binary $(t,t-1)$-Burst Correcting Codes
  14. The First Constraint
  15. The Second Constraint
  16. ...and 9 more sections

Key Result

Theorem 1

Assume $s \neq 0$ and $\boldsymbol{x} \in \Sigma_q^n$, we have $|\mathcal{B}_{t,s}(\boldsymbol{x})| = q^{s-1}((q-1)(n-t+1)+1)$.

Figures (2)

  • Figure 1: Illustration of sequences $\boldsymbol{x}$ and $\boldsymbol{y}$ satisfying Equation (\ref{['eq:z']}) and the condition $j+1\geq i+t$, where an edge indicates that the two symbols are identical.
  • Figure 2: Illustrations of the changes in $\boldsymbol{x}$ and $\alpha(\boldsymbol{x})$ when $\boldsymbol{x}$ suffers a $(t,t-1)$-burst error.

Theorems & Definitions (56)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Corollary 1
  • Remark 2
  • Lemma 1
  • Theorem 3
  • Lemma 2
  • Theorem 4
  • Remark 3
  • ...and 46 more