Sequence Reconstruction Problem for Ternary Deletion Channels

TL;DR

This work solves the sequence reconstruction problem for ternary deletion channels with two deletions per channel by exactly determining $N_3(n,2,t)$. The authors introduce deletion-ball intersection bounds $M_0(n,t)$ and $M_1(n,t)$ based on $D_3(n,t)$, and prove that $N_3(n,2,t)=\max\{M_0(n,t),M_1(n,t)\}$ for $n \ge \max\{6,\lfloor 3t/2\rfloor+1\}$ and $t \ge 2$, with $N_3(n,2,t)=3^{n-t}$ in the regime $\frac{3t}{2} \ge n \ge t$. For small $t$ (2–5), the paper shows $N_3(n,2,t)=M_1(n,t)$ and provides explicit polynomial forms, while a detailed upper-bound analysis confirms the main equality in the broader range. The results advance the understanding of reconstruction from multiple deletion channels and lay groundwork for extending to larger alphabets ($q \ge 4$).

Abstract

The sequence reconstruction problem was proposed by Levenshtein in 2001. In this model, a sequence from a code is transmitted over several channels, and the decoder receives the distinct outputs from each channel. The main problem is to determine the minimum number of channels required to reconstruct the transmitted sequence. In the combinatorial context, the sequence reconstruction problem is equivalent to finding the value of $N_q(n,d,t)$, defined as the size of the largest intersection of two metric balls of radius $t$, where the distance between their centers is at least $d$ and the sequences are $q$-ary sequences of the length $n$. Levenshtein first discussed this problem in the uncoded sequence setting and determined the value of $N_q(n,1,t)$ for any $n\geqslant t$. Moreover, Gabrys and Yaakobi studied this problem in the context of binary one-deletion-correcting codes and determined the value of $N_2(n,2,t)$ for $t\geqslant 2$. In this paper we study this problem for $3$-ary sequences of length $n$ over the deletion channel, where the transmitted sequence belongs to a one-deletion-correcting code and there are $t$ deletions in every channel. Specifically, we determine $N_3(n,2,t)$ for $t\geqslant 2$.

Theorems & Definitions (46)

• Theorem 1: Levenshtein L1
• Theorem 2: Gabrys and Yaakobi Gabrys
• Theorem 3: Pham, Goyal, and Kiah Pham
• Theorem 4
• Lemma 5
• Lemma 6
• Lemma 7
• proof
• Lemma 8
• proof