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Sequence Reconstruction for the Single-Deletion Single-Substitution Channel

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

TL;DR

This work proves that if two q-ary sequences of length n have a Hamming distance and a Levenshtein distance, then the intersection size of their error balls is upper bounded by <inline-formula> and the gap between this bound and the tight bound is at most 2.

Abstract

The central problem in sequence reconstruction is to find the minimum number of distinct channel outputs required to uniquely reconstruct the transmitted sequence. According to Levenshtein's work in 2001, this number is determined by the size of the maximum intersection between the error balls of any two distinct input sequences of the channel. In this work, we study the sequence reconstruction problem for single-deletion single-substitution channel, assuming that the transmitted sequence belongs to a $q$-ary code with minimum Hamming distance at least $2$, where $q\geq 2$ is any fixed integer. Specifically, we prove that for any two $q$-ary sequences of length $n$ and with Hamming distance $d\geq 2$, the size of the intersection of their error balls is upper bounded by $2qn-3q-2-δ_{q,2}$, where $δ_{i,j}$ is the Kronecker delta. We also prove the tightness of this bound by constructing two sequences the intersection size of whose error balls achieves this bound.

Sequence Reconstruction for the Single-Deletion Single-Substitution Channel

TL;DR

This work proves that if two q-ary sequences of length n have a Hamming distance and a Levenshtein distance, then the intersection size of their error balls is upper bounded by <inline-formula> and the gap between this bound and the tight bound is at most 2.

Abstract

The central problem in sequence reconstruction is to find the minimum number of distinct channel outputs required to uniquely reconstruct the transmitted sequence. According to Levenshtein's work in 2001, this number is determined by the size of the maximum intersection between the error balls of any two distinct input sequences of the channel. In this work, we study the sequence reconstruction problem for single-deletion single-substitution channel, assuming that the transmitted sequence belongs to a -ary code with minimum Hamming distance at least , where is any fixed integer. Specifically, we prove that for any two -ary sequences of length and with Hamming distance , the size of the intersection of their error balls is upper bounded by , where is the Kronecker delta. We also prove the tightness of this bound by constructing two sequences the intersection size of whose error balls achieves this bound.
Paper Structure (11 sections, 5 theorems, 90 equations, 6 figures, 1 table)

This paper contains 11 sections, 5 theorems, 90 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem description and main result
  3. Intersection size of error balls of $q$-ary substitution channel
  4. Some useful observations and lemma
  5. The notation $\phi$
  6. Methodology
  7. For $\Lambda^L_0(\bm x,\bm x')$ and $\Lambda^R_0(\bm x,\bm x')$
  8. For $\Lambda^L_1(\bm x,\bm x')$ and $\Lambda^R_1(\bm x,\bm x')$
  9. For $\Lambda^L_2(\bm x,\bm x')$ and $\Lambda^R_2(\bm x,\bm x')$
  10. Proof of Theorem \ref{['thm-ins-size']}
  11. Conclusions and future work

Key Result

Theorem 1

Suppose $n\geq\max\{\frac{q+23}{2}, \frac{5q+19}{q-1}\}$. For any $\bm x,\bm x'\in\Sigma_q^n$ with $d_{\text{H}}(\bm x,\bm x')\geq 2$, we have where $\delta_{i,j}$ is the Kronecker delta. Moreover, there exist two sequences $\bm x,\bm x'\in\Sigma_q^n$ with $d_{\text{H}}(\bm x,\bm x')=2$ and $|B^{\text{D,S}}_{1,1}(\bm x,\bm x')|=2qn-3q-2-\delta_{q,2}$.

Figures (6)

  • Figure 1: An overview of the dividing of $\Lambda$, where $\Lambda=\{(x_{[n]\backslash j}, x'_{[n]\backslash j'}): j,j'\in[n]~\text{and}~ d_{\text{H}}(x_{[n]\backslash j'}, x'_{[n]\backslash j})\leq 2\}$ is defined by \ref{['eq-def-LMD']}. First, $\Lambda$ is divided into $\Lambda_0, \Lambda_1$ and $\Lambda_2$ according to the value of $d_{\text{H}}(x_{[n]\backslash j'}, x'_{[n]\backslash j})$. Then for each $\ell\in\{0,1,2\}$, $\Lambda_\ell$ is divided into $\Lambda_\ell^L$ and $\Lambda_\ell^R$ according to the relationship of $j$ and $j'$. Here we assume $j\leq j'$ and consider $(x_{[n]\backslash j}, x'_{[n]\backslash j'})$ and $(x_{[n]\backslash j'}, x'_{[n]\backslash j})$. Finally, for each $\ell\in\{1,2\}$ and each $X\in\{L,R\}$, $\Lambda_\ell^X$ is divided into $\Lambda_{\ell,i}^X$, $i=1,\cdots,p_\ell$, where $p_1=3$ and $p_2=6$, according to the value of $(|S\cap [1,j-1]|,|T^X\cap [j+1,j']|,|S\cap [j'+1,n]|)$, where by Observation 2, $d_{\text{H}}(x_{[n]\backslash j'}, x'_{[n]\backslash j})=|S\cap [1,j-1]|+|T^X\cap [j+1,j']|+|S\cap [j'+1,n]|$. Moreover, the sets $\Lambda^X_0$ and $\Lambda^X_{\ell,i}$ can be easily obtained from $\bm x$ and $\bm x'$.
  • Figure 2: An illustration of the pair $(j,j')$ satisfying \ref{['eq-LDM0-dst']}. Each black dot represents a symbol of $\bm x$ (in the upper row) or a symbol of $\bm x'$ (in the lower row). Symbols are connected by a solid segment are identical, while those connected by a dashed segment are distinct. Here, $k_a=\max(T^L\cap[1,i_1])$ and $k'_a=n$ because $T^L\cap[i_d+1,n]=\emptyset$. We can find that $(j,j')$ satisfies \ref{['eq-LDM0-dst']} if and only if $k_a\leq j\leq i_1<i_d\leq j'\leq k'_a$. In this example, $d=4$. Moreover, we can find that $x_i=x'_i=x_{i+1}$ for each $i\in [k_a,i_1-1]$ and $x'_i=x_{i}=x'_{i-1}$ for each $i\in [i_d+1,k_a']$. Hence, $x_{[k_a,i_1]}$ is contained in a run of $\bm x$ and $x'_{[i_d,k'_a]}$ is contained in a run of $\bm x'$.
  • Figure 3: An illustration of the pair $(j,j')$ in the proof of Claim 1.1. Here, $S=\{i_1,i_2,i_3,i_4\}$, $k_1$ and $k_1'$ are defined by \ref{['eq-def-k1']} and \ref{['eq-def-k1p']} respectively. According to the proof of Claim 1.1, $j_1=\max(T^L\cap[i_1+1,i_2])$. We can see that $(j,j')$ satisfies the conditions $|S\cap[1,j-1]|=1$ and $|T^L\cap[j+1,j']|=|S\cap[j'+1,n]|=0$ if and only if it satisfies \ref{['eq-dHGE2-LMD1-L2-Clm1']}, that is, $j_1\leq j\leq i_2\leq i_d\leq j'<k'_1$. In fact, we have $S\cap[1,j-1]=S\cap[1,j-1]=\{i_1\}$ and $T^L\cap[j+1,j']=S\cap[j'+1,n]=\emptyset$. Moreover, we can see that $x_{[j_1,i_2]}$ is contained in a run of $\bm x$ and $x'_{[i_d,k'_1-1]}$ is contained in a run of $\bm x'$.
  • Figure 4: An illustration of the pair $(j,j')$ in the proof of Claim 1.2. Here $S=\{i_1,i_2,i_3\}$. In this figure, (a) is for Case (i), (b) is for possibility 1) of Case (ii) and (c) is for possibility 2) of Case (ii). Here, $k_1,k_1',k_b$ and $k_b'$ are defined by \ref{['eq-def-k1']}, \ref{['eq-def-k1p']}, \ref{['defeq-ka-1']} and \ref{['defeq-pka-1']} respectively.
  • Figure 5: An illustration of the pair $(j,j')$ in the proof of Claim 1.3. Here, $S=\{i_1,i_2,i_3\}$ and $j_1' =\min(T^L\cap[i_{2}+1,i_3])$ because $d=3$.
  • ...and 1 more figures

Theorems & Definitions (31)

  • Theorem 1
  • Remark 1
  • Lemma 1
  • proof 1
  • Example 1
  • Remark 2
  • Remark 3
  • Definition 1
  • proof 2
  • Remark 4
  • ...and 21 more