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Binary Reconstruction Codes for Correcting One Deletion and One Substitution

Yuling Li, Yubo Sun, Gennian Ge

TL;DR

This work studies reconstruction codes for channels that introduce one deletion and one substitution, formalized via the single-deletion single-substitution ball $\mathcal{B}$. The authors bound the worst-case intersection $|\mathcal{B}(\bm{x}) \cap \mathcal{B}(\bm{y})|$ by decomposing into deletion and substitution components with parameters $(d,s)\in\{(0,0),(1,0),(2,0),(0,2),(1,2),(2,2)\}$, leveraging $|\mathcal{D}(\bm{x},\bm{y})|=d$ and $|\mathcal{S}(\bm{x},\bm{y})|=s$, and thereby construct reconstruction codes with explicit redundancies for several target $N$. The paper deploys VT-based and run-length constrained schemes, often constrained by the inversion number $\mathrm{Inv}(\cdot)$ modulo small moduli, and, in some cases, combines these with list-decodable constructions. Key results include redundancy $0$ for $N=4n-8$, $1$ for $N=3n-4$, $2$ for $N=2n+9$, $\log\log n+3$ for $N=n+21$, $\log n+1$ for $N=31$, and $3\log n+4$ for $N=7$. These constructions demonstrate explicit binary codes achieving near-optimal reconstruction capabilities with sublinear or logarithmic redundancies, enabling unique or list-decodable reconstructions from a fixed number of noisy outputs. The results advance sequence reconstruction theory for combined deletion and substitution errors and have potential applications in DNA storage and racetrack memory where robust data integrity under multiple error types is essential.

Abstract

In this paper, we investigate binary reconstruction codes capable of correcting one deletion and one substitution. We define the \emph{single-deletion single-substitution ball} function $ \mathcal{B} $ as a mapping from a sequence to the set of sequences that can be derived from it by performing one deletion and one substitution. A binary \emph{$(n,N;\mathcal{B})$-reconstruction code} is defined as a collection of binary sequences of length $ n $ such that the intersection size between the single-deletion single-substitution balls of any two distinct codewords is strictly less than $ N $. This property ensures that each codeword can be uniquely reconstructed from $ N $ distinct elements in its single-deletion single-substitution ball. Our main contribution is to demonstrate that when $ N $ is set to $ 4n - 8 $, $ 3n - 4 $, $2n+9$, $ n+21 $, $31$, and $7$, the redundancy of binary $(n,N;\mathcal{B})$-reconstruction codes can be $0$, $1$, $2$, $ \log\log n + 3 $, $\log n + 1 $, and $ 3\log n + 4 $, respectively, where the logarithm is on base two.

Binary Reconstruction Codes for Correcting One Deletion and One Substitution

TL;DR

This work studies reconstruction codes for channels that introduce one deletion and one substitution, formalized via the single-deletion single-substitution ball . The authors bound the worst-case intersection by decomposing into deletion and substitution components with parameters , leveraging and , and thereby construct reconstruction codes with explicit redundancies for several target . The paper deploys VT-based and run-length constrained schemes, often constrained by the inversion number modulo small moduli, and, in some cases, combines these with list-decodable constructions. Key results include redundancy for , for , for , for , for , and for . These constructions demonstrate explicit binary codes achieving near-optimal reconstruction capabilities with sublinear or logarithmic redundancies, enabling unique or list-decodable reconstructions from a fixed number of noisy outputs. The results advance sequence reconstruction theory for combined deletion and substitution errors and have potential applications in DNA storage and racetrack memory where robust data integrity under multiple error types is essential.

Abstract

In this paper, we investigate binary reconstruction codes capable of correcting one deletion and one substitution. We define the \emph{single-deletion single-substitution ball} function as a mapping from a sequence to the set of sequences that can be derived from it by performing one deletion and one substitution. A binary \emph{-reconstruction code} is defined as a collection of binary sequences of length such that the intersection size between the single-deletion single-substitution balls of any two distinct codewords is strictly less than . This property ensures that each codeword can be uniquely reconstructed from distinct elements in its single-deletion single-substitution ball. Our main contribution is to demonstrate that when is set to , , , , , and , the redundancy of binary -reconstruction codes can be , , , , , and , respectively, where the logarithm is on base two.
Paper Structure (25 sections, 25 theorems, 47 equations, 4 tables)

This paper contains 25 sections, 25 theorems, 47 equations, 4 tables.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Notations
  4. Errors and Codes
  5. Useful Tools
  6. Error Correcting Codes
  7. Reconstruction Codes
  8. Error Ball
  9. Key Lemmas
  10. Reconstruction Codes
  11. The Case of
  12. The Case of
  13. The Case of
  14. The Case of
  15. The Case of
  16. ...and 10 more sections

Key Result

Lemma 2.5

Levenshtein-66-SPD-1D For any $a \in [0,2n-1]$, define $\mathrm{VT}_{a}(n) = \left\{\bm{x} \in \Sigma^n : \mathrm{VT}^{1}(\bm{x}) \equiv a \pmod{2n}\right\}$, it holds that $\mathcal{D}(\bm{x},\bm{y})=\emptyset$ and $\mathcal{S}(\bm{x},\bm{y})=\emptyset$ for any two distinct sequences $\bm{x}, \bm{y

Theorems & Definitions (51)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Remark 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 41 more