Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Sequence Reconstruction Problem for the Single-Deletion Two-Substitution Channel

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

TL;DR

This work analyzes the Levenshtein sequence reconstruction problem for the $q$-ary single-deletion two-substitution channel, where one deletion and up to two substitutions occur per transmission. The authors develop a run-based combinatorial framework to bound the intersection size of error balls $|B^{DS}_{1,2}(\bm x,\bm x')|$ for two length-$n$ sequences with $d_{H}(\bm x,\bm x')\ge 2$, proving the upper bound $|B^{DS}_{1,2}(\bm x,\bm x')| \le (q^2-1)n^2-(3q^2+5q-5)n+O_q(1)$ and showing tightness up to an additive constant. The analysis proceeds by splitting into the $d_{H}=2$ case and the $d_{H}\ge 3$ case, and further into subcases governed by run structure and the relative positions of deletions and substitutions; auxiliary results for substitution-ball sizes and their intersections are used to control the combinatorics. The result yields a precise asymptotic characterization of error-ball intersections, informing the design of reconstruction codes and their read coverage under mixed error models with one deletion and multiple substitutions. The bound can be made fully explicit in the constant term, and the overall approach provides a robust template for mixed-error reconstruction problems with similar channel models.

Abstract

The Levenshtein sequence reconstruction problem studies the reconstruction of a transmitted sequence from multiple erroneous copies of it. A fundamental question in this field is to determine the minimum number of erroneous copies required to guarantee correct reconstruction of the original sequence. This problem is equivalent to determining the maximum possible intersection size of two error balls associated with the underlying channel. Existing research on the sequence reconstruction problem has largely focused on channels with a single type of error, such as insertions, deletions, or substitutions alone. However, relatively little is known for channels that involve a mixture of error types, for instance, channels allowing both deletions and substitutions. In this work, we study the sequence reconstruction problem for the single-deletion two-substitution channel, which allows one deletion and at most two substitutions applied to the transmitted sequence. Specifically, we prove that if two $q$-ary length-$n$ sequences have the Hamming distance $d\geq 2$, where $q\geq 2$ is any fixed integer, then the intersection size of their error balls under the single-deletion two-substitution channel is upper bounded by $(q^2-1)n^2-(3q^2+5q-5)n+O_q(1)$, where $O_q(1)$ is a constant independent from $n$ but dependent on $q$. Moreover, we show that this upper bound is tight up to an additive constant.

On the Sequence Reconstruction Problem for the Single-Deletion Two-Substitution Channel

TL;DR

This work analyzes the Levenshtein sequence reconstruction problem for the -ary single-deletion two-substitution channel, where one deletion and up to two substitutions occur per transmission. The authors develop a run-based combinatorial framework to bound the intersection size of error balls for two length- sequences with , proving the upper bound and showing tightness up to an additive constant. The analysis proceeds by splitting into the case and the case, and further into subcases governed by run structure and the relative positions of deletions and substitutions; auxiliary results for substitution-ball sizes and their intersections are used to control the combinatorics. The result yields a precise asymptotic characterization of error-ball intersections, informing the design of reconstruction codes and their read coverage under mixed error models with one deletion and multiple substitutions. The bound can be made fully explicit in the constant term, and the overall approach provides a robust template for mixed-error reconstruction problems with similar channel models.

Abstract

The Levenshtein sequence reconstruction problem studies the reconstruction of a transmitted sequence from multiple erroneous copies of it. A fundamental question in this field is to determine the minimum number of erroneous copies required to guarantee correct reconstruction of the original sequence. This problem is equivalent to determining the maximum possible intersection size of two error balls associated with the underlying channel. Existing research on the sequence reconstruction problem has largely focused on channels with a single type of error, such as insertions, deletions, or substitutions alone. However, relatively little is known for channels that involve a mixture of error types, for instance, channels allowing both deletions and substitutions. In this work, we study the sequence reconstruction problem for the single-deletion two-substitution channel, which allows one deletion and at most two substitutions applied to the transmitted sequence. Specifically, we prove that if two -ary length- sequences have the Hamming distance , where is any fixed integer, then the intersection size of their error balls under the single-deletion two-substitution channel is upper bounded by , where is a constant independent from but dependent on . Moreover, we show that this upper bound is tight up to an additive constant.
Paper Structure (18 sections, 6 theorems, 59 equations, 5 tables)

This paper contains 18 sections, 6 theorems, 59 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Intersection of error balls for substitution channel
  4. Some observations and more auxiliary results
  5. Intersection of error balls of sequences with Hamming distance $d=2$
  6. For the case that both $x_{[n]\backslash\{i_1\}} =x'_{[n]\backslash\{i_2\}}$ and $x_{[n]\backslash\{i_2\}} =x'_{[n]\backslash\{i_1\}}$
  7. For the case that either $x_{[n]\backslash\{i_1\}} \neq x'_{[n]\backslash\{i_2\}}$ or $x_{[n]\backslash\{i_2\}} \neq x'_{[n]\backslash\{i_1\}}$ but not both
  8. For the case that both $x_{[n]\backslash\{i_1\}} \neq x'_{[n]\backslash\{i_2\}}$ and $x_{[n]\backslash\{i_2\}}\neq x'_{[n]\backslash\{i_1\}}$
  9. Intersection of error balls of sequences with Hamming distance $d\geq 3$
  10. Intersection of error balls of sequences with Hamming distance $d=3$
  11. Intersection of error balls of sequences with Hamming distance $d=4$
  12. Intersection of error balls of sequences with Hamming distance $d\geq 5$
  13. Proof of Claims 1$-$5
  14. Proof of Claim 1
  15. Proof of Claim 2
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose $\bm x,\bm x'\in\Sigma_q^n$ such that $d_{\text{H}}(\bm x,\bm x')\geq 2$. We have $|B^{\text{DS}}_{1,2}(\bm x, \bm x')|\leq (q^2-1)n^2-(3q^2+5q-5)n+O_q(1)$, where $O_q(1)$ is a constant independent of $n$ but dependent on $q$. Moreover, there exist two sequences $\bm x,\bm x'\in\Sigma_q^n$ s

Theorems & Definitions (22)

  • Theorem 1
  • Lemma 1
  • proof 1
  • Lemma 2
  • proof 2
  • Remark 1
  • Lemma 3
  • proof 3
  • Lemma 4
  • proof 4
  • ...and 12 more